{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 263 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 264 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 265 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 264 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 264 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE " " -1 270 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "The integral test and the p - ser ies" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can ada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 17 "The integral test" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Suppose that the te rms of the series " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$Sum G6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 15 " have the form " }{XPPEDIT 18 0 "a[n]=f(n)" "6#/&%\"aG6#%\"nG-%\"fG6#F'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 47 " is a c ontinuous decreasing function such that " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 24 " is greater than 0 when " }{TEXT 279 1 "x" }{TEXT -1 19 " is greater than 0." }}{PARA 0 "" 0 "" {TEXT -1 35 "Then the sequence of partial sums " }{XPPEDIT 18 0 "S[n] = Sum(a[i],i = 1 .. n);" "6#/&%\"SG6#%\"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" }{TEXT -1 32 " and the sequence of integrals " }{XPPEDIT 18 0 "I[n] = Int(f( x),x=1..n)" "6#/&%\"IG6#%\"nG-%$IntG6$-%\"fG6#%\"xG/F.;\"\"\"F'" } {TEXT -1 13 " are either " }{TEXT 263 15 "both convergent" }{TEXT -1 4 " or " }{TEXT 263 14 "both divergent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "In the case that the sequence " }{XPPEDIT 18 0 "I[n] " "6#&%\"IG6#%\"nG" }{TEXT -1 29 " converges to a finite limit " } {XPPEDIT 18 0 "I" "6#%\"IG" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity) <= I+a[1];" "6#1-%$SumG6$&%\"aG6#%\"nG/F* ;\"\"\"%)infinityG,&%\"IGF-&F(6#F-F-" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 266 34 "__________________________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Limit(S[n],n = infinity);" "6#-%&LimitG6$&%\"SG6#%\" nG/F)%)infinityG" }{TEXT -1 6 ", is " }{XPPEDIT 18 0 "Sum(a[n],n = 1 \+ .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "Limit(I[n],n=infinity)" "6#-%&LimitG6$ &%\"IG6#%\"nG/F)%)infinityG" }{TEXT -1 16 " is denoted by " } {XPPEDIT 18 0 "Int(f(x),x=1..infinity)" "6#-%$IntG6$-%\"fG6#%\"xG/F); \"\"\"%)infinityG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 21 "A s eries of the form " }{XPPEDIT 18 0 "Sum(a[n],n = N .. infinity)" "6#-% $SumG6$&%\"aG6#%\"nG/F);%\"NG%)infinityG" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "N >= 2" "6#1\"\"#%\"NG" }{TEXT -1 90 ", can be tested f or convergence in a similar way by considering the sequence of integra ls " }{XPPEDIT 18 0 "I[n]=Int(f(x),x=N..n)" "6#/&%\"IG6#%\"nG-%$IntG6$ -%\"fG6#%\"xG/F.;%\"NGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 424 336 336 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"3%**************>\"!#<$\"3qLLLLLLL$) !#=7$$\"3cmmmJ?.h7F*$\"3]cZ?zE,IzF-7$$\"3PLL3wc898F*$\"33E)\\iNk&4wF-7 $$\"3cmm;yi&QP\"F*$\"35d3iY7hm\"F*$\"3ycu>vn*>+' F-7$$\"3#)***\\(pGKD%) Q)f&F-7$$\"3smm;$*Q()R=F*$\"3$=?RU6b^V&F-7$$\"3x*****44m-!>F*$\"3Q^!oP !4Ui_F-7$$\"3w*****\\E14'>F*$\"3)3yhW#Go*4&F-7$$\"31+++4VM>?F*$\"3MgZ_ *\\-@&\\F-7$$\"3immT^;Ts?F*$\"33S?&z;'HD[F-7$$\"3TLLLaQ^N@F*$\"3w=GK3P r#o%F-7$$\"3YLLL(pp*)=#F*$\"31\\')3m#f$oXF-7$$\"3_***\\2b^6D#F*$\"3(Gp w)R41BF*$\"3A2-%*y/:OVF-7$$\"3i***\\2D#emBF*$\"3)p(p &3$G]DUF-7$$\"3')***\\A2(3CCF*$\"3](Ql!eVEDTF-7$$\"3'HLL3K(3%[#F*$\"33 W^K\"QBc-%F-7$$\"3FLL3zi=RDF*$\"3Z%G_\"[&p#QRF-7$$\"3amm;\\zh)f#F*$\"3 sNJP(Q*>[QF-7$$\"3WmmT^2NgEF*$\"3O`%=>%G!*ePF-7$$\"3!****\\sL*39FF*$\" 3[d!\\h4xWo$F-7$$\"3_mm;#[G@x#F*$\"3N)\\/vSOtg$F-7$$\"3X*****zf)3KGF*$ \"3lKsa\")H'4`$F-7$$\"3i****\\tyu!*GF*$\"39b&GCT7$fMF-7$$\"3/++vQP]ZHF *$\"3k\"H<$z8q#R$F-7$$\"3v****\\]9_5IF*$\"31Vp#)*o$o@LF-7$$\"3gKLLnc9n IF*$\"3u7OS5.OgKF-7$$\"3k*****\\5/w7$F*$\"3)3!ou;cL(>$F-7$$\"3EmmTN%)Q #=$F*$\"3sX&))G(RHUJF-7$$\"3/+++z@GUKF*$\"3OE!*f#GZU3$F-7$$\"3aKL3Cvj) H$F*$\"3-*\\(zsYbJIF-7$$\"35++v(4XvN$F*$\"3!y\\Vsam$yHF-7$$\"3'GLL$=!Q ^T$F*$\"3[xW(\\MR\"GHF-7$$\"3M***\\(oCVvMF*$\"3GWVl_(Rt(GF-7$$\"3'pmm1 >.N`$F*$\"3jYdgIE0IGF-7$$\"35mm;\"*)))Gf$F*$\"3Co5P/cF$y#F-7$$\"3?LL3[ Gy^OF*$\"3!3SmtX)QQFF-7$$\"3m*****zF+fq$F*$\"3InxYY'*R)p#F-7$$\"3aLL$) f\\#zw$F*$\"3e$3.vm!)Rl#F-7$$\"3hmmmu0SBQF*$\"3g3(yfusah#F-7$$\"31++v] \"\\D)QF*$\"3'H5jlSFcd#F-7$$\"3))***\\_)=;RRF*$\"3^7v,y5hQDF-7$$\"\"% \"\"!$\"3++++++++DF--%'COLOURG6&%$RGBG$\"*++++\"!\")$FfzFfzF`[l-%*THIC KNESSG6#\"\"#-%)POLYGONSG6$7&7$$Fd[lFfzF`[l7$$\"\"$FfzF`[l7$F\\\\l$\"+ LLLLL!#57$Fj[lF_\\l-%&COLORG6&F\\[l$\"\"'!\"\"$\"\")Fh\\lFf\\l-Ff[l6$7 &Fb\\lF^\\l7$F\\\\l$\"+++++]Fa\\l7$Fj[lF_]l-Fd\\l6&F\\[l$\"#&*!\"#$\" \"(Fh\\lFi\\l-%%TEXTG6%7$$\"#;Fh\\l$\"#xFf]lQ)y~=~f(x)6\"Fiz-Fj]l6%7$F j[l$!\"$Ff]lQ\"nFb^l-Fjz6&F\\[lFfzFfzFfz-Fj]l6%7$F\\\\lFf^lQ$n+1Fb^lFi ^l-Fj]l6%7$Fdz$Ff]lFf]lQ\"xFb^lFi^l-Fj]l6%7$$\"#ZFf]lFd]lQ\"yFb^lFi^l- %*AXESTICKSG6$FfzFfz-%+AXESLABELSG6$Fc_lQ!Fb^l-%%VIEWG6$;$FezFh\\lFdz; $!\"&Ff]l$\"\"\"Ffz" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 58 "The green rectangle in the picture has width 1 and height " } {XPPEDIT 18 0 "f(n+1)=a[n+1]" "6#/-%\"fG6#,&%\"nG\"\"\"F)F)&%\"aG6#,&F (F)F)F)" }{TEXT -1 25 ", so its area represents " }{XPPEDIT 18 0 "a[n+ 1]" "6#&%\"aG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "The rectangle consisting of the red and green parts has w idth 1 and height " }{XPPEDIT 18 0 "f(n)=a[n]" "6#/-%\"fG6#%\"nG&%\"aG 6#F'" }{TEXT -1 25 ", so its area represents " }{XPPEDIT 18 0 "a[n]" " 6#&%\"aG6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "The pi cture shows that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " a[n+1] < Int(f(x),x=n..n+1)" "6#2&%\"aG6#,&%\"nG\"\"\"F)F)-%$IntG6$-% \"fG6#%\"xG/F0;F(,&F(F)F)F)" }{XPPEDIT 18 0 "`` < a[n]" "6#2%!G&%\"aG6 #%\"nG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " } {XPPEDIT 18 0 "n >= 1" "6#1\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "Adding the inequalites " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[2] < Int(f(x),x = 1 .. 2);" "6#2&%\"aG6#\" \"#-%$IntG6$-%\"fG6#%\"xG/F.;\"\"\"F'" }{XPPEDIT 18 0 "`` < a[1];" "6# 2%!G&%\"aG6#\"\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[3] < Int(f(x),x = 2 .. 3);" "6#2&%\"aG6#\"\"$-%$In tG6$-%\"fG6#%\"xG/F.;\"\"#F'" }{XPPEDIT 18 0 "`` < a[2];" "6#2%!G&%\"a G6#\"\"#" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 2 " :" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n] < Int(f(x),x = n -1 .. n);" "6#2&%\"aG6#%\"nG-%$IntG6$-%\"fG6#%\"xG/F.;,&F'\"\"\"F2!\" \"F'" }{XPPEDIT 18 0 "`` < a[n-1];" "6#2%!G&%\"aG6#,&%\"nG\"\"\"F*!\" \"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "gives" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[2]+a[3]" "6#,&&%\"aG6#\"\"#\" \"\"&F%6#\"\"$F(" }{TEXT -1 11 " + . . . + " }{XPPEDIT 18 0 "a[n] < In t(f(x),x = 1 .. n);" "6#2&%\"aG6#%\"nG-%$IntG6$-%\"fG6#%\"xG/F.;\"\"\" F'" }{XPPEDIT 18 0 "``LqF*$\"3i]SHSXS#>\"F-7$$\"3Mm;z%>y&\\wF *$\"3+vM^AaNV6F-7$$\"3)**\\(oaPwU$)F*$\"3U.\"[+\"e#[4\"F-7$$\"3hLLe9$ \\f.*F*$\"3OC]HAO*>0\"F-7$$\"3RL$3--^J/\"F-$\"3a&>8#ya)4z*F*7$$\"3cm\" HK#Q/#=\"F-$\"3qMOs(4%y(>*F*7$$\"3?Le9J_\"3J\"F-$\"3t/68WKKM()F*7$$\"3 (**\\7`c]TW\"F-$\"3(G2?c\\\\8K)F*7$$\"3?Lekeh/#e\"F-$\"330]sekU]zF*7$$ \"3)**\\i!>&*\\>()3'=F-$\"3-`W,ewgItF* 7$$\"3IL$3x`@a)>F-$\"3!zF#*o)3)p4(F*7$$\"3K++D6xhD@F-$\"3o2mhsl%*eoF*7 $$\"3!****\\Pa*QmAF-$\"3FYmV6x^UmF*7$$\"3z***\\()G\\?S#F-$\"31Lvp_\"=A X'F*7$$\"3.LekL8CDDF-$\"3#eLSO.nGH'F*7$$\"3ammT!fG$eF* 7$$\"3Mmm;*4[z1$F-$\"3Z*G(46(4#4dF*7$$\"3\")*\\7yOP\"3KF-$\"3?H=esd2$e 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YGONSG6'7&7$$\"\"\"Fe[lF_\\l7$$Fc\\lFe[lF_\\l7$F\\]l$\"+5y1rq!#57$Fi\\ lF^]l7&F[]l7$$\"\"$Fe[lF_\\l7$Fd]l$\"+$p-Nx&F`]l7$F\\]lFg]l7&7$$\"\"%F e[lF_\\l7$$\"\"&Fe[lF_\\l7$F_^l$\"+af8sWF`]l7$F\\^lFb^l7&F^^l7$$\"\"'F e[lF_\\l7$Fg^l$\"+1H[#3%F`]l7$F_^lFj^l-%&COLORG6&F[\\l$Fh^l!\"\"$\"\") Fa_lF`_l-Fe\\l6'7&Fa]lF]]l7$F\\]lFi\\l7$Fi\\lFi\\l7&Fi]lFf]l7$Fd]lF^]l F]]l7&Fd^lFa^l7$F_^l$\"+++++]F`]l7$F\\^lF]`l7&F\\_lFi^l7$Fg^lFb^lFa^l- F^_l6&F[\\l$\"#&*!\"#$Fd[lFa_lFb_l-%%TEXTG6%7$$\"#6Fa_l$\"$D\"Ff`lQ)y~ =~f(x)6\"Fh[l-Fi`l6%7$$\"#NFa_l$\"#DFf`lQ$...Faal-Fi[l6&F[\\lFe[lFe[lF e[l-Fi`l6%7$Fi\\l$!\"'Ff`lQ\"1FaalFjal-Fi`l6%7$F\\]lF_blQ\"2FaalFjal-F i`l6%7$Fd]lF_blQ\"3FaalFjal-Fi`l6%7$F\\^lF_blQ$n-2FaalFjal-Fi`l6%7$F_^ lF_blQ$n-1FaalFjal-Fi`l6%7$Fg^lF_blQ\"nFaalFjal-Fi`l6%7$Fc[l$!\"%Ff`lQ \"xFaalFjal-Fi`l6%7$$Fa_lFa_l$\"$X\"Ff`lQ\"yFaalFjal-%*AXESTICKSG6$Fe[ lFe[l-%+AXESLABELSG6$F[dlQ!Faal-%%VIEWG6$;F_dlFc[l;F_bl$\"#:Fa_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The total green area represents " }{XPPEDIT 18 0 "S[n]-a[1]" "6#,&&%\"SG6#%\"nG \"\"\"&%\"aG6#F(!\"\"" }{TEXT -1 45 ", the combined red and green area represents " }{XPPEDIT 18 0 "S[n-1]=S[n]-a[n]" "6#/&%\"SG6#,&%\"nG\" \"\"F)!\"\",&&F%6#F(F)&%\"aG6#F(F*" }{TEXT -1 31 ", and the area under the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "x=n" "6#/%\"xG%\"nG" }{TEXT -1 12 " represents " }{XPPEDIT 18 0 "Int(f(x),x = 1 .. n)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\" \"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Now suppose \+ that the sequence " }{XPPEDIT 18 0 "I[n]" "6#&%\"IG6#%\"nG" }{TEXT -1 25 " tends to a finite limit " }{TEXT 273 1 "I" }{TEXT -1 4 " as " }{TEXT 272 1 "n" }{TEXT -1 19 " tends to infinity." }}{PARA 0 "" 0 "" {TEXT -1 21 "Then from (i) we have" }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "S[n]-a[1] < I[n]" "6#2,&&%\"SG6#%\"nG\"\"\"&%\"aG6#F )!\"\"&%\"IG6#F(" }{XPPEDIT 18 0 "``<= I" "6#1%!G%\"IG" }{TEXT -1 11 " for all " }{TEXT 274 1 "n" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "n > = 1" "6#1\"\"\"%\"nG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] < I[n]+a[1 ];" "6#2&%\"SG6#%\"nG,&&%\"IG6#F'\"\"\"&%\"aG6#F,F," }{XPPEDIT 18 0 "` ` <= I+a[1];" "6#1%!G,&%\"IG\"\"\"&%\"aG6#F'F'" }{TEXT -1 9 " for all \+ " }{TEXT 275 1 "n" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "n >= 1" "6#1\" \"\"%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "It follows \+ that the sequence of partial sums " }{XPPEDIT 18 0 "S[n]=Sum(a[i],i=1. .n)" "6#/&%\"SG6#%\"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" }{TEXT -1 22 " converges to a limit " }{XPPEDIT 18 0 "S=Sum(a[n],n=1..infinity) " "6#/%\"SG-%$SumG6$&%\"aG6#%\"nG/F+;\"\"\"%)infinityG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "S<=I+a[1]" "6#1%\"SG,&%\"IG\"\"\"&%\"aG6#F'F' " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Similarly, if the seq uence " }{XPPEDIT 18 0 "S[n]" "6#&%\"SG6#%\"nG" }{TEXT -1 25 " tends t o a finite limit " }{XPPEDIT 18 0 "S = Sum(a[n],n = 1 .. infinity)" "6 #/%\"SG-%$SumG6$&%\"aG6#%\"nG/F+;\"\"\"%)infinityG" }{TEXT -1 4 " as \+ " }{TEXT 277 1 "n" }{TEXT -1 32 " tends to infinity, (i) gives: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[n]infinity" "6#f *6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "I[n]+a[n]infinity" "6#f*6#%\"nG7\" 6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 17 ", the inequal ity " }{XPPEDIT 18 0 "S[n]-a[1] " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "p>0" "6#2\"\"!%\"pG" }{TEXT -1 6 ", the " }{TEXT 263 8 "p-series" } {TEXT -1 6 " " }{XPPEDIT 18 0 "Sum(1/(n^p),n = 1 .. infinity) = 1 +1/(2^p)+1/(3^p)+1/(4^p)+1/(5^p)+` . . . `;" "6#/-%$SumG6$*&\"\"\"F()% \"nG%\"pG!\"\"/F*;F(%)infinityG,.F(F(*&F(F()\"\"#F+F,F(*&F(F()\"\"$F+F ,F(*&F(F()\"\"%F+F,F(*&F(F()\"\"&F+F,F(%(~.~.~.~GF(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 32 " " } {TEXT 265 1 "I" }{TEXT -1 41 " - converges to a sum which is less than " }{XPPEDIT 18 0 "p/(p-1)" "6#*&%\"pG\"\"\",&F$F%F%!\"\"F'" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "p > 1" "6#2\"\"\"%\"pG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 32 " " } {TEXT 265 2 "II" }{TEXT -1 29 " - diverges to infinity when " } {XPPEDIT 18 0 "p <= 1" "6#1%\"pG\"\"\"" }{TEXT -1 1 "." }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{TEXT 267 24 "________________________" }{TEXT -1 29 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The special cases " }{XPPEDIT 18 0 "p=1 " "6#/%\"pG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p=2" "6#/%\"pG \"\"#" }{TEXT -1 47 " are considered separately in other worksheets." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "p = 1" "6#/%\"pG\" \"\"" }{TEXT -1 22 ", the p-series is the " }{TEXT 263 15 "harmonic se ries" }{TEXT -1 37 ", which diverges to infinity, and if " }{XPPEDIT 18 0 "p=2" "6#/%\"pG\"\"#" }{TEXT -1 26 ", the series converges to " } {XPPEDIT 18 0 "Pi^2/6" "6#*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 54 "Not that, in all cases, the terms tend to 0, that is, " }{XPPEDIT 18 0 "Limit(1/n^p,p=infinity)=0" "6#/-%&Limit G6$*&\"\"\"F()%\"nG%\"pG!\"\"/F+%)infinityG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "I[n]=Int(1/x^p,x=1. .n)" "6#/&%\"IG6#%\"nG-%$IntG6$*&\"\"\"F,)%\"xG%\"pG!\"\"/F.;F,F'" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(x^(-p),x=1..n)" "6#-%$IntG6$)%\"x G,$%\"pG!\"\"/F';\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Then , if " }{XPPEDIT 18 0 "p<>1" "6#0%\"pG\"\"\"" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I[n]= x^(-p+1) " "6#/&%\"IG6#%\"nG)%\"xG,&%\"pG!\"\"\"\"\"F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([n, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$%\"nG% !G7$\"\"\"F(" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "n^(-p+1)/(-p+1)-1/(-p+1)" "6#,&*&)%\"nG,&%\"pG!\"\"\"\" \"F*F*,&F(F)F*F*F)F**&F*F*,&F(F)F*F*F)F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 4 "= " }{XPPEDIT 18 0 "1/(p-1)-n^(1-p)/(p-1);" "6#, &*&\"\"\"F%,&%\"pGF%F%!\"\"F(F%*&)%\"nG,&F%F%F'F(F%,&F'F%F%F(F(F(" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I[n] = (1-n^(1-p))/(p-1);" "6#/&%\"IG6#%\"nG*&,&\"\"\"F*)F',&F*F*%\"pG!\" \"F.F*,&F-F*F*F.F." }{TEXT -1 6 " or " }{XPPEDIT 18 0 "I[n] = 1/(p-1 );" "6#/&%\"IG6#%\"nG*&\"\"\"F),&%\"pGF)F)!\"\"F," }{TEXT -1 2 " " } {XPPEDIT 18 0 "``(1-1/(n^(p-1)));" "6#-%!G6#,&\"\"\"F'*&F'F')%\"nG,&% \"pGF'F'!\"\"F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "p>1" "6#2\"\"\"%\"pG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1/n^(p-1)" "6#*&\"\"\"F$)%\"nG,&%\"pGF$F$!\"\"F)" }{TEXT -1 15 " tend s to 0 as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG% &arrowG6\"%)infinityGF*F*F*" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "I[n] " "6#&%\"IG6#%\"nG" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "1/(p-1) " "6#*&\"\"\"F$,&%\"pGF$F$!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Then, by the " }{TEXT 263 13 "integral test" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(1/n^p,n=1..infinity)" "6#-%$SumG6$*&\"\"\"F') %\"nG%\"pG!\"\"/F);F'%)infinityG" }{TEXT -1 20 " converges to a sum " }{TEXT 259 1 "S" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "S<=1+1/(p-1)" "6 #1%\"SG,&\"\"\"F&*&F&F&,&%\"pGF&F&!\"\"F*F&" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "p/(p-1)" "6#*&%\"pG\"\"\",&F$F%F%!\"\"F'" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "p<1" "6#2%\"pG\" \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n^(1-p)" "6#)%\"nG,&\"\"\"F&%\" pG!\"\"" }{TEXT -1 23 " tends to infinity, as " }{XPPEDIT 18 0 "n->inf inity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "I[n]" "6#&%\"IG6#%\"nG" }{TEXT -1 19 " tends to infinity." }}{PARA 0 "" 0 "" {TEXT -1 13 "Then, by the \+ " }{TEXT 263 13 "integral test" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Sum(1 /(n^p),n = 1 .. infinity)" "6#-%$SumG6$*&\"\"\"F')%\"nG%\"pG!\"\"/F);F '%)infinityG" }{TEXT -1 22 " diverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "p=1" "6#/%\"pG\"\"\"" }{TEXT -1 83 ", the p - series is the harmonic series, which is handled in a differ ent worksheet." }}{PARA 0 "" 0 "" {TEXT -1 57 "This case can also be d ealt with using the integral test." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "p=1" "6#/%\"pG\"\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[n]=Int(1/x,x=1..n)" "6#/&%\"I G6#%\"nG-%$IntG6$*&\"\"\"F,%\"xG!\"\"/F-;F,F'" }{TEXT -1 2 " " }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = ln(x);" "6#/%!G -%#lnG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([n, ``],[1, ` `]);" "6#-%*PIECEWISEG6$7$%\"nG%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(n);" "6#/%!G-%#lnG6 #%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence the sequ ence " }{XPPEDIT 18 0 "I[n]" "6#&%\"IG6#%\"nG" }{TEXT -1 25 " diverges to infinity as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)oper atorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 263 13 "integral test" }{TEXT -1 12 " shows t hat " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity)" "6#-%$SumG6$*&\"\"\" F'%\"nG!\"\"/F(;F'%)infinityG" }{TEXT -1 22 " diverges to infinity." } }{PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 20 ": Once we have that \+ " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity)" "6#-%$SumG6$*&\"\"\"F'% \"nG!\"\"/F(;F'%)infinityG" }{TEXT -1 72 " diverges to infinity, the \+ comparison test could be used to show that " }{XPPEDIT 18 0 "Sum(1/n^ p,n = 1 .. infinity)" "6#-%$SumG6$*&\"\"\"F')%\"nG%\"pG!\"\"/F);F'%)in finityG" }{TEXT -1 26 " diverges to infinity for " }{XPPEDIT 18 0 "p<1 " "6#2%\"pG\"\"\"" }{TEXT -1 8 ", since " }{XPPEDIT 18 0 "1/n^p>1/n" " 6#2*&\"\"\"F%%\"nG!\"\"*&F%F%)F&%\"pGF'" }{TEXT -1 6 " for " } {XPPEDIT 18 0 "p < 1" "6#2%\"pG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n>=1" "6#1\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT 258 8 "Question" }{TEXT 268 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "Sum(1/(n*sqrt(n)),n=1..infi nity)" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'-%%sqrtG6#F)F'!\"\"/F);F'%)infin ityG" }{TEXT -1 54 " converges, and find an approximate value for the \+ sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 265 8 "Solution" }{TEXT 269 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "The series has the form of a p - series " }{XPPEDIT 18 0 "Sum(1/n^p,n=1..infinit y)" "6#-%$SumG6$*&\"\"\"F')%\"nG%\"pG!\"\"/F);F'%)infinityG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "p=3/2" "6#/%\"pG*&\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "p>1" "6#2\"\"\"%\"pG" }{TEXT -1 23 ", the series converges." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Sum(1/(n*sqrt(n)),n=1..infinity);\nvalue(%);\nS := evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$)%\"nG#\"\"$\"\"# F'!\"\"/F*;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%ZetaG6# #\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG$\"+\\`P7E!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The con vergence is rather slow as indicated by the following picture." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "pts := [seq([n,sum(1/i^1.5,i=1..n)],n=1..40)]:\nplot([pts,Zeta(1. 5),pts],x=0..40,style=[line$2,point],\n color=[gray,cyan,blue],lines tyle=[2,3],symbol=circle,\n view=[0..40,0..3],labels=[`n`,``]);" }} {PARA 13 "" 1 "" {GLPLOT2D 486 258 258 {PLOTDATA 2 "6(-%'CURVESG6&7J7$ $\"\"\"\"\"!F(7$$\"\"#F*$\"33+++!R`NN\"!#<7$$\"\"$F*$\"3)*******zM+Y:F 07$$\"\"%F*$\"3)*******zM+r;F07$$\"\"&F*$\"3++++*>Y/w\"F07$$\"\"'F*$\" 34+++\"e([G=F07$$\"\"(F*$\"35+++1D[#)=F07$$\"\")F*$\"3#*******zmnE>F07 $$\"\"*F*$\"33+++F07$$\"#5F*$\"3-+++%\\O`*>F07$$\"#6F*$\"30+++;mu A?F07$$\"#7F*$\"3=+++xG!o/#F07$$\"#8F*$\"3=++++v8o?F07$$\"#9F*$\"3#)** ****)eFs3#F07$$\"#:F*$\"3')*****z%3W/@F07$$\"#;F*$\"3()*****z%e1?@F07$ $\"#F*$\"3?+++J< ]f@F07$$\"#?F*$\"3))*****42#oq@F07$$\"#@F*$\"3=+++*Rt5=#F07$$\"#AF*$\" 3%******4Mk2>#F07$$\"#BF*$\"39+++&=I)*>#F07$$\"#CF*$\"3!)*****zNN$3AF0 7$$\"#DF*$\"3*)******e`L;AF07$$\"#EF*$\"3z*****pGyQA#F07$$\"#FF*$\"37+ ++og+JAF07$$\"#GF*$\"3))*****RVbxB#F07$$\"#HF*$\"3*)*****>seTC#F07$$\" #IF*$\"33+++GXC]AF07$$\"#JF*$\"3A+++Z#QgD#F07$$\"#KF*$\"3%*******=DchA F07$$\"#LF*$\"3/+++*fPoE#F07$$\"#MF*$\"3%******\\n\")=F#F07$$\"#NF*$\" 33+++G6rwAF07$$\"#OF*$\"3!)*****zvS8G#F07$$\"#PF*$\"35+++uRy&G#F07$$\" #QF*$\"31+++fH0!H#F07$$\"#RF*$\"3!)******3)eTH#F07$$\"#SF*$\"35+++c;6) H#F0-%'COLOURG6&%$RGBG$\")=THv!\")F]xF]x-%&STYLEG6#%%LINEG-%*LINESTYLE G6#F--F$6&7S7$$F*F*$\"37+++\\`P7EF07$$\"3Hmmmm;')=()!#=F\\y7$$\"3RLLLe '40j\"F0F\\y7$$\"3ommm;6m$[#F0F\\y7$$\"3Dnmm;yYULF0F\\y7$$\"3ELLLeF>(> %F0F\\y7$$\"3%omm;>K'*)\\F0F\\y7$$\"3g*****\\Kd,\"eF0F\\y7$$\"3Onmm\"f X(emF0F\\y7$$\"3O+++DCh/vF0F\\y7$$\"3uMLLL/pu$)F0F\\y7$$\"3mnmm;c0T\"* F0F\\y7$$\"35+++I,Q+5!#;F\\y7$$\"31+++]*3q3\"Fc[lF\\y7$$\"3/+++q=\\q6F c[lF\\y7$$\"3umm;fBIY7Fc[lF\\y7$$\"3TLLLj$[kL\"Fc[lF\\y7$$\"3ZLLL`Q\"G T\"Fc[lF\\y7$$\"3.++]s]k,:Fc[lF\\y7$$\"3GLLL`dF!e\"Fc[lF\\y7$$\"3G++]s gam;Fc[lF\\y7$$\"3;++]Fc[lF\\y7$$\"3kmmmTc-)*>Fc[lF\\y7$$\"3)omm\"f`@'3#Fc[lF\\y7$$\"31 ++]nZ)H;#Fc[lF\\y7$$\"3+nmmJy*eC#Fc[lF\\y7$$\"3/+++S^bJBFc[lF\\y7$$\"3 7+++0TN:CFc[lF\\y7$$\"3A++]7RV'\\#Fc[lF\\y7$$\"3++++:#fke#Fc[lF\\y7$$ \"31LLL`4NnEFc[lF\\y7$$\"3?+++],s`FFc[lF\\y7$$\"3\\mm;zM)>$GFc[lF\\y7$ $\"3Z+++qfa " 0 "" {MPLTEXT 1 0 51 "Sum(1/(i*sqrt(i)),i=1..10000000);\nevalf(evalhf(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$)%\"iG#\"\"$\"\"#F'!\"\"/F* ;F'\")+++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$*Gu6E!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT 258 8 "Question" }{TEXT 270 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that \+ the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n = 2 .. infinity);" "6#- %$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\"\"#%)infinityG" } {TEXT -1 30 " diverges and that the series " }{XPPEDIT 18 0 "Sum(1/(n* ln(n)^2),n = 2 .. infinity)" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'*$-%#lnG6# F)\"\"#F'!\"\"/F);F.%)infinityG" }{TEXT -1 12 " converges. " }}{PARA 0 "" 0 "" {TEXT -1 59 "Find an approximate value for the sum of the se cond series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 265 8 "Solution" }{TEXT 271 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[ln(ln(x))]=1/(x*ln(x))" "6#/7#-%#lnG6#-F&6#%\"xG*& \"\"\"F,*&F*F,-F&6#F*F,!\"\"" }{TEXT -1 3 " so" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[n]" "6#&%\"IG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(1/(x*ln(x)),x=2..n) = ln(ln(x)" "6#/-%$IntG6$*& \"\"\"F(*&%\"xGF(-%#lnG6#F*F(!\"\"/F*;\"\"#%\"nG-F,6#-F,6#F*" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "PIECEWISE([n, ``],[2, ``]) = ln(ln(n))-ln(ln (2));" "6#/-%*PIECEWISEG6$7$%\"nG%!G7$\"\"#F),&-%#lnG6#-F.6#F(\"\"\"-F .6#-F.6#F+!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus \+ " }{XPPEDIT 18 0 "Limit(I[n],n=infinity)=infinity" "6#/-%&LimitG6$&%\" IG6#%\"nG/F*%)infinityGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "It follows that the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n = 2 .. infinity)" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\" \"#%)infinityG" }{TEXT -1 44 " diverges to infinity by the integral te st.\023" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int(1/(x*ln(x)),x=2..n);\nvalue(%);\nLimit(%,n=infini ty);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F' *&%\"xGF'-%#lnG6#F)F'!\"\"/F);\"\"#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#-F%6#%\"nG\"\"\"-F%6#-F%6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,&-%#lnG6#-F(6#%\"nG\"\"\"-F(6#-F(6#\"\"# !\"\"/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 22 ": Although \+ the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n = 2 .. infinity)" "6#-% $SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\"\"#%)infinityG" } {TEXT -1 42 " diverges to infinity, it is nevertheless " }{TEXT 263 17 "very slow growing" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 " Now consider the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)^2),n = 2 .. i nfinity)" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'*$-%#lnG6#F)\"\"#F'!\"\"/F);F .%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[-1/ln(x)] = 1/(x*ln(x)^2);" "6#/7#,$*&\"\"\"F'-%#lnG6# %\"xG!\"\"F,*&F'F'*&F+F'*$-F)6#F+\"\"#F'F," }{TEXT -1 3 " so" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[n]" "6#&%\"IG6#%\"nG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(1/(x*ln(x)^2),x = 2 .. n) = -1/ln (x);" "6#/-%$IntG6$*&\"\"\"F(*&%\"xGF(*$-%#lnG6#F*\"\"#F(!\"\"/F*;F/% \"nG,$*&F(F(-F-6#F*F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([n, ``],[2, ``]) = -1/ln(n)+1/ln(2);" "6#/-%*PIECEWISEG6$7$%\"nG%!G7$\"\" #F),&*&\"\"\"F.-%#lnG6#F(!\"\"F2*&F.F.-F06#F+F2F." }{TEXT -1 3 " . " } }{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "Limit(I[n],n = inf inity) = 1/ln(2);" "6#/-%&LimitG6$&%\"IG6#%\"nG/F*%)infinityG*&\"\"\"F .-%#lnG6#\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 260 1 "~" }{TEXT -1 12 "1. 442695041." }}{PARA 0 "" 0 "" {TEXT -1 27 "It follows that the series \+ " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)^2),n = 2 .. infinity);" "6#-%$SumG6$* &\"\"\"F'*&%\"nGF'*$-%#lnG6#F)\"\"#F'!\"\"/F);F.%)infinityG" }{TEXT -1 32 " converges by the integral test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\023" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Int(1/(x*ln(x)^2),x=2..n);\nvalue(%);\nLimit(%,n=infi nity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\" F'*&%\"xGF')-%#lnG6#F)\"\"#F'!\"\"/F);F.%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%#lnG6#\"\"#!\"\"-F&6#%\"nG\"\"\"F-*&F*F-F%F-F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&-%#lnG6#\"\"#!\"\"-F)6 #%\"nG\"\"\"F0*&F-F0F(F0F,/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$-%#lnG6#\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "The series converges to a sum which is n o greater than " }{XPPEDIT 18 0 "1/ln(2)+1/(2*ln(2)^2)" "6#,&*&\"\"\" F%-%#lnG6#\"\"#!\"\"F%*&F%F%*&F)F%*$-F'6#F)F)F%F*F%" }{TEXT -1 1 " " } {TEXT 261 1 "~" }{TEXT -1 13 " 2.483379532." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The convergence is rather slow \+ as indicated by the following picture." }}{PARA 0 "" 0 "" {TEXT -1 62 "(The approximation 2.1097428 for the sum is calculated below.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "PS := n -> sum(1/(i*ln(i)^2),i=2..n):\npts := [seq([n,evalhf(PS(n ))],n=2..50)]:\nplot([pts,2.1097428,pts],x=0..50,style=[line$2,point], \n color=[gray,cyan,blue],linestyle=[2,3],symbol=circle,\n view=[0 ..50,0..2.5],labels=[`n`,``]);" }}{PARA 13 "" 1 "" {GLPLOT2D 483 253 253 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$\"\"#\"\"!$\"3(Q!G]!\\%oS5!#<7$$\" \"$F*$\"33yGttH'oJ\"F-7$$\"\"%F*$\"3_Gd/N&[pW\"F-7$$\"\"&F*$\"3*ycE_&* fT_\"F-7$$\"\"'F*$\"3W0[!Q`ugd\"F-7$$\"\"(F*$\"3(zGY=*=!Qh\"F-7$$\"\") F*$\"3ma-O%z4Fk\"F-7$$\"\"*F*$\"3'3wizmCdm\"F-7$$\"#5F*$\"3AARmPee%o\" F-7$$\"#6F*$\"3D)GFXQ'R+F*$\"3=PL*z?5Jx\"F-7$$\"#?F*$\"375ft.;oyCrz\"F-7$$\"#DF*$\"3dol.hZ)4!=F-7$$\"#EF*$\"3h\"HF[,3Y!=F-7$ $\"#FF*$\"3rm8tDw,3=F-7$$\"#GF*$\"3]-\"o&*3M7\"=F-7$$\"#HF*$\"3eE1)3Dv U\"=F-7$$\"#IF*$\"38V;KFn:<=F-7$$\"#JF*$\"31%>SSD#*)>=F-7$$\"#KF*$\"38 ^kElR\\A=F-7$$\"#LF*$\"3DM)phis\\#=F-7$$\"#MF*$\"3-3t-DyLF=F-7$$\"#NF* $\"3a*HvM8)fH=F-7$$\"#OF*$\"3_fg\"4Ch<$=F-7$$\"#PF*$\"32bn&y1MQ$=F-7$$ \"#QF*$\"3]>*p>'G#e$=F-7$$\"#RF*$\"3\"3)***)zKtP=F-7$$\"#SF*$\"3$[ZYsX q&R=F-7$$\"#TF*$\"3@W\"*oo!R8%=F-7$$\"#UF*$\"32%4`)zL/V=F-7$$\"#VF*$\" 3P[3A#H(oW=F-7$$\"#WF*$\"33Cb;#Qui%=F-7$$\"#XF*$\"31+D*\\$z!y%=F-7$$\" #YF*$\"3sxgsu4H\\=F-7$$\"#ZF*$\"3A_^D!HE2&=F-7$$\"#[F*$\"3C#*>Nek6_=F- 7$$\"#\\F*$\"3yIj*R'QY`=F-7$$\"#]F*$\"3mJ&p$=2xa=F--%'COLOURG6&%$RGBG$ \")=THv!\")F\\[lF\\[l-%&STYLEG6#%%LINEG-%*LINESTYLEG6#F)-F$6&7S7$$F*F* $\"3\")*******zU(4@F-7$$\"3SLLL3x&)*3\"F-F[\\l7$$\"3zmm\"H2P\"Q?F-F[\\ l7$$\"3XLL$eRwX5$F-F[\\l7$$\"3=ML$3x%3yTF-F[\\l7$$\"3gmm\"z%4\\Y_F-F[ \\l7$$\"34LLeR-/PiF-F[\\l7$$\"3;++DcmpisF-F[\\l7$$\"3vLLe*)>VB$)F-F[\\ l7$$\"3o++DJbw!Q*F-F[\\l7$$\"3%ommTIOo/\"!#;F[\\l7$$\"3^LL3_>jU6F[^lF[ \\l7$$\"3E++]i^Z]7F[^lF[\\l7$$\"3/++](=h(e8F[^lF[\\l7$$\"3A++]P[6j9F[^ lF[\\l7$$\"3[L$e*[z(yb\"F[^lF[\\l7$$\"3+nm;a/cq;F[^lF[\\l7$$\"3mmmm;t, mF[^lF[\\l7$$\"3M+]i !f#=$3#F[^lF[\\l7$$\"37+](=xpe=#F[^lF[\\l7$$\"3-nm\"H28IH#F[^lF[\\l7$$ \"3%om\"zpSS\"R#F[^lF[\\l7$$\"3cLL3_?`(\\#F[^lF[\\l7$$\"3fL$e*)>pxg#F[ ^lF[\\l7$$\"3D+]Pf4t.FF[^lF[\\l7$$\"3ZLLe*Gst!GF[^lF[\\l7$$\"39+++DRW9 HF[^lF[\\l7$$\"3:++DJE>>IF[^lF[\\l7$$\"35+]i!RU07$F[^lF[\\l7$$\"3$)*** \\(=S2LKF[^lF[\\l7$$\"3nmmm\"p)=MLF[^lF[\\l7$$\"3U++](=]@W$F[^lF[\\l7$ $\"36L$e*[$z*RNF[^lF[\\l7$$\"3e++]iC$pk$F[^lF[\\l7$$\"3Sm;H2qcZPF[^lF[ \\l7$$\"3Y+]7.\"fF&QF[^lF[\\l7$$\"3amm;/OgbRF[^lF[\\l7$$\"3I+]ilAFjSF[ ^lF[\\l7$$\"3)RLLL)*pp;%F[^lF[\\l7$$\"3WLL3xe,tUF[^lF[\\l7$$\"3Wn;HdO= yVF[^lF[\\l7$$\"3a+++D>#[Z%F[^lF[\\l7$$\"3)om;aG!e&e%F[^lF[\\l7$$\"3wL LL$)Qk%o%F[^lF[\\l7$$\"3m+]iSjE!z%F[^lF[\\l7$$\"3u+]P40O\"*[F[^lF[\\l7 $FdzF[\\l-Fiz6&F[[lFj[l$\"*++++\"F^[lF^elF_[l-Fd[l6#F0-F$6&F&-Fiz6&F[[ lFj[lFj[lF^el-F`[l6#%&POINTGFc[l-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$% \"nG%!G-%%VIEWG6$;Fj[lFdz;Fj[l$Fhr!\"\"" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " The part ial sums with " }{XPPEDIT 18 0 "10^m" "6#)\"#5%\"mG" }{TEXT -1 14 " te rms, where " }{TEXT 262 1 "m" }{TEXT -1 46 " is between 1 and 7 inclus ive are as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "PS := n -> Sum(1/(i*ln(i)^2),i=2..n);\ne valf(evalhf(PS(10)));\nevalf(evalhf(PS(100)));\nevalf(evalhf(PS(1000)) );\nevalf(evalhf(PS(10000)));\nevalf(evalhf(PS(100000)));\nevalf(evalh f(PS(1000000)));\nevalf(evalhf(PS(10000000)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PSGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&\" \"\"F0*&%\"iGF0)-%#lnG6#F2\"\"#F0!\"\"/F2;F79$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Qee%o\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+h2$G*=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]%))\\'>!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q(p6+#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VR)G-#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+! Rgt.#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+L2qZ?!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The calculation in the following subsection shows that the sum of " }{XPPEDIT 18 0 "1 0^(10^8)" "6#)\"#5*$F$\"\")" }{TEXT -1 34 " terms is approximately 2.1 097428." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Estimation \+ of partial sums" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "The following code sets up a function EM, using the Eule r-Maclaurin summation formula, which can be used to calculate approxim ations for the partial sums" }{XPPEDIT 18 0 "Sum(1/(i*ln(i)^2),i = 2 . . n);" "6#-%$SumG6$*&\"\"\"F'*&%\"iGF'*$-%#lnG6#F)\"\"#F'!\"\"/F);F.% \"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "k := 25:\neulermac(1/((j+k)*ln(j+k)^2),j= 2-k..n-k):\nEM := unapply(convert(%,polynom),n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#EMGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,bp*&#\"\"\"\" ++Dc,TF/*&F/F/*$)-%#lnG6#\"#D\"\"'F/!\"\"F/F/*&#\"$*p\"+++voaF/*&F/F/* $)F4\"\"%F/F9F/F9*&#\"%L_\",+vVB:'F/*&F/F/*$)F4\"\"&F/F9F/F9*&#\"+F6Mx C\"-+](o/B\"F/*&F/F/*$)F4\"\"#F/F9F/F/*&#F/\",]i!RD5F/*&F/F/*$)F4\"\"( F/F9F/F/*&#\"#6\"$g$F/*&F/F/*&)9$FAF/)-F56#Fjn\"\"$F/F9F/F/*&#\"*J;-Y# \"-+Dc^G#*F/*&F/F/*$)F4F^oF/F9F/F/*&#F/\"$?\"F/*&F/F/*&FinF/)F\\oFQF/F 9F/F/*&#F/\"#7F/*&F/F/*&)FjnFQF/F[pF/F9F/F9*&#F/\"$_#F/*&F/F/*&)FjnF8F /F[pF/F9F/F9*&#\"$P\"\"%gvF/*&F/F/*&FgpF/F[oF/F9F/F9*&#FI\"$7\"F/*&F/F /*&FgpF/)F\\oFAF/F9F/F9*&#\"#F/ *&F/F/*$)-F56#FfyFQF/F9F/F/*&#F/FfqF/*&F/F/*$)-F56#FfqFQF/F9F/F/*&#F/ \"#=F/*&F/F/*$)-F56#FezFQF/F9F/F/*&#F/\"#9F/*&F/F/*$)-F56#F][lFQF/F9F/ F/*&#F/\"#:F/*&F/F/*$)-F56#Fe[lFQF/F9F/F/*&#F/\"#;F/*&F/F/*$)-F56#F]\\ lFQF/F9F/F/*&#F/F^pF/*&F/F/*$)-F56#F^pFQF/F9F/F/*&#F/\"#8F/*&F/F/*$)-F 56#F\\]lFQF/F9F/F/*&FhvF/*&F/F/*&FjnF/F[pF/F9F/F/*&#F/\"#UF/*&F/F/*&Fg pF/)F\\oFXF/F9F/F9*&#F/\"#IF/*&F/F/*&FinF/FiqF/F9F/F/*&#F/F8F/*&F/F/*& FapF/F[oF/F9F/F9*&#F/FQF/*(,&*&FQF/FguF/F/F\\oF9F/F\\oF9FguF9F/F9F(F(F (" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Her e are some examples for comparison." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Sum(1/(i*ln(i)^2),i=2..1000 )=\nevalf(evalhf(sum(1/(i*ln(i)^2),i=2..1000)));\nevalf(EM(1000));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%\"iGF()-%#lnG6# F*\"\"#F(!\"\"/F*;F/\"%+5$\"+]%))\\'>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^%))\\'>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Sum(1/(i*ln(i)^2),i=2..1000 0)=\n evalf(evalhf(sum(1/(i*ln(i)^2),i=2..10000)));\nevalf(EM (10000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%\" iGF()-%#lnG6#F*\"\"#F(!\"\"/F*;F/\"&++\"$\"+q(p6+#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r(p6+#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Sum(1/(i*ln(i)^2),i=2..100 000)=\n evalf(evalhf(sum(1/(i*ln(i)^2),i=2..100000)));\nevalf (EM(100000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(* &%\"iGF()-%#lnG6#F*\"\"#F(!\"\"/F*;F/\"'++5$\"+VR)G-#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+WR)G-#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Sum(1/(i*ln(i)^2),i=2.. 1000000)=\n evalf(evalhf(sum(1/(i*ln(i)^2),i=2..1000000)));\n evalf(EM(1000000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\" \"\"F(*&%\"iGF()-%#lnG6#F*\"\"#F(!\"\"/F*;F/\"(+++\"$\"+!Rgt.#!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"Rgt.#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Sum(1/(i*ln (i)^2),i=2..10000000)=\n evalf(evalhf(sum(1/(i*ln(i)^2),i=2.. 10000000)));\nevalf(EM(10000000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$SumG6$*&\"\"\"F(*&%\"iGF()-%#lnG6#F*\"\"#F(!\"\"/F*;F/\")+++5$\"+L 2qZ?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N2qZ?!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Here are some fur ther calculations of partial sums of " }{XPPEDIT 18 0 "10^(10^m)" "6#) \"#5)F$%\"mG" }{TEXT -1 73 " terms for m = 1 to 8, using only the Eule r-Maclaurin summation formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(evalf(EM(Float(1,10^m))) ,m=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*$\"+aLJm?!\"*$\"+d)*R0@F% $\"+3&3$4@F%$\"+t$*p4@F%$\"+f%Q(4@F%$\"+oBu4@F%$\"+fFu4@F%$\"+)zU(4@F% " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "evalf(EM(1e100000000));\nevalf(EM(1e200000000));\nev alf(EM(1e300000000));\nevalf(EM(8.537682564e357913923));\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)zU(4@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++Gu4@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,Gu4@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,Gu4@!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "Sum(1/(s qrt(n)*ln(n)),n = 2 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*&-%%sqrtG6#% \"nGF'-%#lnG6#F,F'!\"\"/F,;\"\"#%)infinityG" }{TEXT -1 22 " diverges t o infinity." }}{PARA 0 "" 0 "" {TEXT -1 45 "__________________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the seri es " }{XPPEDIT 18 0 "Sum(ln(n)/(n^2),n = 1 .. infinity);" "6#-%$SumG6$ *&-%#lnG6#%\"nG\"\"\"*$F*\"\"#!\"\"/F*;F+%)infinityG" }{TEXT -1 54 " c onverges, and find an approximate value for the sum." }}{PARA 0 "" 0 " " {TEXT -1 45 "_____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "__ ___________________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 21 "Show that the series " }{XPPEDIT 18 0 "Su m(1/(n*ln(n)^3),n = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'*$ -%#lnG6#F)\"\"$F'!\"\"/F);F'%)infinityG" }{TEXT -1 54 " converges, and find an approximate value for the sum." }}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 20 "Code for 1st picture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "p1:= plot(1/x,x=1.2..4,color=red,thickness=2):\np2 : = plots[polygonplot]([[2,0],[3,0],[3,1/3],[2,1/3]],\n color=COLOR(RG B,.6,.8,.6)):\np3 := plots[polygonplot]([[2,1/3],[3,1/3],[3,1/2],[2,1/ 2]],\n color=COLOR(RGB,.95,.7,.8)):\nt1 := plots[textplot]([1.6,.77, `y = f(x)`],color=red):\nt2 := plots[textplot]([[2,-.03,`n`],[3,-.03,` n+1`],\n [4,-.02,`x`],[0.47,.95,`y`]],color=black):\nplots[display]( [p1,p2,p3,t1,t2],view=[0.4..4,-0.05..1],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "Code for 2nd picture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 785 "s := x -> 1/sqrt(x):\ny2:=s(2): y3 :=s(3): y4:=s(4): y5 :=s(5): y6 :=s(6):\np1:=plot(s(x),x=0.5..7,color= red,thickness=2):\np2:=plots[polygonplot]([[[1,0],[2,0],[2,y2],[1,y2]] ,\n [[2,0],[3,0],[3,y3],[2,y3]],[[4,0],[5,0],[5,y5],[4,y5]],\n [[5 ,0],[6,0],[6,y6],[5,y6]]],color=COLOR(RGB,.6,.8,.6)):\np3:=plots[polyg onplot]([[[1,y2],[2,y2],[2,1],[1,1]],\n [[2,y3],[3,y3],[3,y2],[2,y2] ],[[4,y5],[5,y5],[5,y4],[4,y4]],\n [[5,y6],[6,y6],[6,y5],[5,y5]]],co lor=COLOR(RGB,.95,.7,.8)):\nt1:=plots[textplot]([1.1,1.25,`y = f(x)`], color=red):\nt2:=plots[textplot]([[3.5,0.25,`...`],[1,-0.06,`1`],\n \+ [2,-0.06,`2`],[3,-0.06,`3`],[4,-0.06,`n-2`],[5,-0.06,`n-1`],\n [6,-0 .06,`n`],[7,-0.04,`x`],[-0.1,1.45,`y`]],color=black):\nplots[display]( [p1,p2,p3,t1,t2],view=[-0.1..7,-0.06..1.5],\n tickmarks=[0,0]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }