{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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 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 "" 260 276 "" 1 18 0 0 0 0 0 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 Output " -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 Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 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 34 "Cesaro summation of Fourier serie s" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{PARA 0 "" 0 "" {TEXT -1 20 "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 30 "The Cesaro limit of a sequence" }{TEXT 262 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "If there are problems with the convergence of a sequence " } {XPPEDIT 18 0 "a[1], a[2]" "6$&%\"aG6#\"\"\"&F$6#\"\"#" }{TEXT -1 39 " , . . . then it can be replaced by the " }{TEXT 261 20 "sequence of av erages" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1],(a[1]+a[2])/2,(a[1]+a[2]+a[3])/3,` . . . `,1/n;" "6'&%\"aG6 #\"\"\"*&,&&F$6#F&F&&F$6#\"\"#F&F&F-!\"\"*&,(&F$6#F&F&&F$6#F-F&&F$6#\" \"$F&F&F7F.%(~.~.~.~G*&F&F&%\"nGF." }{XPPEDIT 18 0 "Sum(a[k],k = 1 .. \+ n),` . . . `;" "6$-%$SumG6$&%\"aG6#%\"kG/F);\"\"\"%\"nG%(~.~.~.~G" } {TEXT -1 12 " ------ (i)" }}{PARA 0 "" 0 "" {TEXT -1 65 "This is a re asonable thing to do because of the following result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "If a sequence " } {XPPEDIT 18 0 "a[1],a[2],` . . . `;" "6%&%\"aG6#\"\"\"&F$6#\"\"#%(~.~. ~.~G" }{TEXT -1 22 " converges to a limit " }{TEXT 265 1 "L" }{TEXT -1 11 ", then the " }{TEXT 261 20 "sequence of averages" }{TEXT -1 23 " (i) also converges to " }{TEXT 266 1 "L" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 25 "_________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "If the sequence of averages converges, we call its limit \+ the " }{TEXT 261 12 "Cesaro limit" }{TEXT -1 26 " of the original sequ ence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Explanation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "b[n] = 1/n;" "6#/&%\"bG6#%\"nG*&\"\"\"F)F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k],k = 1 .. n);" "6#-%$SumG6$&%\" aG6#%\"kG/F);\"\"\"%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "epsilon" "6#% (epsilonG" }{TEXT -1 35 " be a (small) positive real number." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "a[n] -> L " "6#f*6#&%\"a G6#%\"nG7\"6$%)operatorG%&arrowG6\"%\"LGF-F-F-" }{TEXT -1 33 ", we can find a positive integer " }{TEXT 267 1 "N" }{TEXT -1 15 " (depending \+ on " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 12 ") such that " }{XPPEDIT 18 0 "abs(a[n]-L)N" "6#2%\"NG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "D = Sum(abs(a[k]-L),k = 1 .. N);" "6 #/%\"DG-%$SumG6$-%$absG6#,&&%\"aG6#%\"kG\"\"\"%\"LG!\"\"/F/;F0%\"NG" } {TEXT -1 14 " , and choose " }{TEXT 268 1 "M" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "M >=N" "6#1%\"NG%\"MG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "M>2*D/epsilon" "6#2*(\"\"#\"\"\"%\"DGF&%(epsilonG!\"\"% \"MG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Then, if " } {XPPEDIT 18 0 "n > M" "6#2%\"MG%\"nG" }{TEXT -1 9 ", we have" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(b[n]-L) = 1/n;" "6#/-%$absG6#,&&%\"bG6#%\"nG\"\"\"%\"LG!\"\" *&F,F,F+F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Sum(a[k],k = 1 .. n)-n *L);" "6#-%$absG6#,&-%$SumG6$&%\"aG6#%\"kG/F-;\"\"\"%\"nGF0*&F1F0%\"LG F0!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/n;" "6#/%!G*&\"\"\"F&%\"n G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Sum(a[k],k = 1 .. n)-Sum(L ,k = 1 .. n) )" "6#-%$absG6#,&-%$SumG6$&%\"aG6#%\"kG/F-;\"\"\"%\"nGF0- F(6$%\"LG/F-;F0F1!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/n;" "6#/%! G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Sum(a[k]-L, k = 1 .. n));" "6#-%$absG6#-%$SumG6$,&&%\"aG6#%\"kG\"\"\"%\"LG!\"\"/F- ;F.%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/n;" "6#1%!G*&\"\"\"F&% \"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(a[k]-L),k = 1 .. n) ;" "6#-%$SumG6$-%$absG6#,&&%\"aG6#%\"kG\"\"\"%\"LG!\"\"/F-;F.%\"nG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/n;" "6#/%!G*&\"\"\"F&%\"nG!\"\" " }{XPPEDIT 18 0 "``(Sum(abs(a[k]-L),k = 1 .. N)+Sum(abs(a[k]-L),k = N +1 .. n));" "6#-%!G6#,&-%$SumG6$-%$absG6#,&&%\"aG6#%\"kG\"\"\"%\"LG!\" \"/F1;F2%\"NGF2-F(6$-F+6#,&&F/6#F1F2F3F4/F1;,&F7F2F2F2%\"nGF2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/n;" "6#1%!G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(D+(n-N)*``(epsilon/2));" "6#-%!G6#,&%\"DG \"\"\"*&,&%\"nGF(%\"NG!\"\"F(-F$6#*&%(epsilonGF(\"\"#F-F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/n;" "6#1%!G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(M*``(epsilon/2)+(n-N)*``(epsilon/2));" "6# -%!G6#,&*&%\"MG\"\"\"-F$6#*&%(epsilonGF)\"\"#!\"\"F)F)*&,&%\"nGF)%\"NG F/F)-F$6#*&F-F)F.F/F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/n;" "6#/% !G*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[M+n-N]*``(eps ilon/2);" "6#*&7#,(%\"MG\"\"\"%\"nGF'%\"NG!\"\"F'-%!G6#*&%(epsilonGF' \"\"#F*F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/n;" "6#1%!G*&\"\"\" F&%\"nG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[n+n]*``(epsilon/2);" "6 #*&7#,&%\"nG\"\"\"F&F'F'-%!G6#*&%(epsilonGF'\"\"#!\"\"F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Examples of sequences which c onverge in the ordinary sense" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 " " 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[k]=1/2^k" "6#/&%\"aG6#%\"kG *&\"\"\"F))\"\"#F'!\"\"" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "Limit( a[k],k = infinity) = 0;" "6#/-%&LimitG6$&%\"aG6#%\"kG/F*%)infinityG\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[n] = 1/n" "6#/&%\"bG6#%\"nG*&\"\"\"F)F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[k],k=1..n) = 1/n" "6#/-%$SumG6$&%\"aG6#%\"kG/F* ;\"\"\"%\"nG*&F-F-F.!\"\"" }{XPPEDIT 18 0 "``(1-1/(2^n));" "6#-%!G6#,& \"\"\"F'*&F'F')\"\"#%\"nG!\"\"F," }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Limit(b[n],n=infinity)=0" "6#/-%&LimitG6$&%\"bG6#%\"nG/F*%)infinity G\"\"!" }{TEXT -1 7 " also. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum(1/2^k,k=1..n)/n;\nsimpli fy(value(%));\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$*&\"\"\"F()\"\"#%\"kG!\"\"/F+;F(%\"nGF(F/F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&)\"\"#,$%\"nG!\"\"\"\"\"F+F* F+F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&)\"\"#,$% \"nG!\"\"\"\"\"F.F-F.F,F-F-/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "a[k]= 1-2/(k*(k+1))" "6#/&%\"aG6#%\"kG,&\"\"\"F)*&\"\"#F)*&F'F),&F'F)F)F)F)! \"\"F." }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "Limit(a[k],k = infinity ) = 1;" "6#/-%&LimitG6$&%\"aG6#%\"kG/F*%)infinityG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n] = 1/n" "6#/& %\"bG6#%\"nG*&\"\"\"F)F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k ],k = 1 .. n) = (n-1)/(n+1);" "6#/-%$SumG6$&%\"aG6#%\"kG/F*;\"\"\"%\"n G*&,&F.F-F-!\"\"F-,&F.F-F-F-F1" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Limit(b[n],n = infinity) = 1;" "6#/-%&LimitG6$&%\"bG6#%\"nG/F*%)infin ityG\"\"\"" }{TEXT -1 6 " also." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Sum(1-2/(k*(k+1)),k=1..n)/n; \nnormal(value(%));\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$,&\"\"\"F(*&F(F(*&%\"kGF(,&F+F(F(F(F(!\" \"!\"#/F+;F(%\"nGF(F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"nG\" \"\"!\"\"F&F&,&F%F&F&F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG 6$*&,&%\"nG\"\"\"!\"\"F)F),&F(F)F)F)F*/F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Examples o f divergent sequences which have a Cesaro limit " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a[k] = (-1)^k;" "6#/&%\"aG6#%\"kG) ,$\"\"\"!\"\"F'" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "Limit(a[k],k = infinity);" "6#-%&LimitG6$&%\"aG6#%\"kG/F)%)infinityG" }{TEXT -1 17 " does not exist." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[ n] = 1/n" "6#/&%\"bG6#%\"nG*&\"\"\"F)F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[k],k = 1 .. n) = ((-1)^n-1)/(2*n);" "6#/-%$SumG6$ &%\"aG6#%\"kG/F*;\"\"\"%\"nG*&,&),$F-!\"\"F.F-F-F3F-*&\"\"#F-F.F-F3" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "Limit(b[n],n=infinity)=0" "6#/-% &LimitG6$&%\"bG6#%\"nG/F*%)infinityG\"\"!" }{TEXT -1 33 ", that is, th e Cesaro limit is 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Sum((-1)^k,k=1..n)/n;\nsimplify(value(%)) ;\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&-%$SumG6$)!\"\"%\"kG/F);\"\"\"%\"nGF,F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&)!\"\"%\"nG\"\"\"F'F)F)F(F'#F)\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,&)!\"\"%\"nG\"\"\"F*F,F,F+F*#F, \"\"#/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{XPPEDIT 18 0 "a[k] = sin(k*Pi/4)^2;" "6#/&%\"aG6#%\"kG*$-%$sinG6 #*(F'\"\"\"%#PiGF-\"\"%!\"\"\"\"#" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "Limit(a[k],k = infinity);" "6#-%&LimitG6$&%\"aG6#%\"kG/F)%)infin ityG" }{TEXT -1 17 " does not exist." }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[n] = 1/n" "6#/&%\"bG6#%\"nG*&\"\"\"F)F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k],k = 1 .. n) = 1/(4*n);" "6#/-% $SumG6$&%\"aG6#%\"kG/F*;\"\"\"%\"nG*&F-F-*&\"\"%F-F.F-!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "[2*n+1-sin(Pi*n/2)-cos(Pi*n/2)]" "6#7#,**&\" \"#\"\"\"%\"nGF'F'F'F'-%$sinG6#*(%#PiGF'F(F'F&!\"\"F.-%$cosG6#*(F-F'F( F'F&F.F." }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(b[n],n = infinity ) = 1/2;" "6#/-%&LimitG6$&%\"bG6#%\"nG/F*%)infinityG*&\"\"\"F.\"\"#!\" \"" }{TEXT -1 31 ", that is, the Cesaro limit is " }{XPPEDIT 18 0 "1/2 " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Sum(sin(k*Pi/4)^2,k=1 ..n)/n;\ncombine(value(%));\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$*$)-%$sinG6#,$*&%\"kG\"\"\"%#PiGF /#F/\"\"%\"\"#F//F.;F/%\"nGF/F6!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&,*-%$cosG6#,$*&%#PiG\"\"\"%\"nGF+#F+\"\"##!\"\"\"\"%F,F--%$sinGF'F /#F+F1F+F+F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,*-%$co sG6#,$*&%#PiG\"\"\"%\"nGF.#F.\"\"##!\"\"\"\"%F/F0-%$sinGF*F2#F.F4F.F.F /F3/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}} {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 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Cesaro summability of a series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 28 "The convergence \+ of a series " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$SumG6$&% \"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 10 " to a sum " }{TEXT 269 1 "S" }{TEXT -1 72 " is defined in terms of the convergence of the sequence of partial sums " }{XPPEDIT 18 0 "S[n]=Sum(a[k],k=1..n)" "6# /&%\"SG6#%\"nG-%$SumG6$&%\"aG6#%\"kG/F.;\"\"\"F'" }{TEXT -1 4 " to " } {TEXT 270 1 "S" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "If the \+ Cesaro limit of the sequence of partial sums " }{XPPEDIT 18 0 "S[n]" " 6#&%\"SG6#%\"nG" }{TEXT -1 26 " exists, it is called the " }{TEXT 261 11 "Cesaro sum " }{TEXT -1 14 "of the series " }{XPPEDIT 18 0 "Sum(a[n ],n = 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "If the series converges in the ordinary sense, then the Cesaro sum coincides with the ordinar y sum." }}{PARA 0 "" 0 "" {TEXT -1 32 "The sequence of partial sums is :" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S[1] = a[1]" "6#/ &%\"SG6#\"\"\"&%\"aG6#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S[2] = a[1]+a[2]" "6#/&%\"SG6#\"\"#,&&%\"aG6#\" \"\"F,&F*6#F'F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "S[3] = a[1]+a[2]+a[3]" "6#/&%\"SG6#\"\"$,(&%\"aG6#\"\" \"F,&F*6#\"\"#F,&F*6#F'F," }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "S[4] = a[1]+a[2]+a[3]+a[4]" "6#/&%\"SG6#\"\" %,*&%\"aG6#\"\"\"F,&F*6#\"\"#F,&F*6#\"\"$F,&F*6#F'F," }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 " : " }}{PARA 0 "" 0 "" {TEXT -1 13 " : " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "S[n] = a[1]+a[2]+` . . . `+a[n]" "6#/&%\"SG6#%\"nG,*&% \"aG6#\"\"\"F,&F*6#\"\"#F,%(~.~.~.~GF,&F*6#F'F," }{TEXT -1 3 ", " }} {PARA 0 "" 0 "" {TEXT -1 12 " :" }}{PARA 0 "" 0 "" {TEXT -1 33 "so the corresponding sequence of " }{TEXT 261 12 "Cesaro means" } {TEXT -1 4 " is " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[ 1]" "6#&%\"MG6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "S[1] = a[1]" "6#/&%\"SG6#\"\"\"&%\"aG6#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[2] = (S[1]+S[2])/2;" "6#/&%\"MG6#\"\"#*&,& &%\"SG6#\"\"\"F-&F+6#F'F-F-F'!\"\"" }{XPPEDIT 18 0 " ``= (2*a[1]+a[2] )/2" "6#/%!G*&,&*&\"\"#\"\"\"&%\"aG6#F)F)F)&F+6#F(F)F)F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[3] = (S[1 ]+S[2]+S[3])/3;" "6#/&%\"MG6#\"\"$*&,(&%\"SG6#\"\"\"F-&F+6#\"\"#F-&F+6 #F'F-F-F'!\"\"" }{XPPEDIT 18 0 " ``= (3*a[1]+2*a[2]+a[3])/3" "6#/%!G*& ,(*&\"\"$\"\"\"&%\"aG6#F)F)F)*&\"\"#F)&F+6#F.F)F)&F+6#F(F)F)F(!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[4] \+ = (S[1]+S[2]+S[3]+S[4])/4;" "6#/&%\"MG6#\"\"%*&,*&%\"SG6#\"\"\"F-&F+6# \"\"#F-&F+6#\"\"$F-&F+6#F'F-F-F'!\"\"" }{XPPEDIT 18 0 "`` = (4*a[1]+3* a[2]+2*a[3]+a[4])/4" "6#/%!G*&,**&\"\"%\"\"\"&%\"aG6#F)F)F)*&\"\"$F)&F +6#\"\"#F)F)*&F1F)&F+6#F.F)F)&F+6#F(F)F)F(!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 14 " : " }}{PARA 0 "" 0 "" {TEXT -1 15 " : " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[n] = (S[1]+S[2]+` . . . `+S[n])/n;" "6#/&%\"MG6#%\"nG*&,*&%\"S G6#\"\"\"F-&F+6#\"\"#F-%(~.~.~.~GF-&F+6#F'F-F-F'!\"\"" }{XPPEDIT 18 0 " ``= (n*a[1]+(n-1)*a[2]+(n-2)*a[3]+` . . . `+2*a[n-1]+a[n])/n" "6#/%! G*&,.*&%\"nG\"\"\"&%\"aG6#F)F)F)*&,&F(F)F)!\"\"F)&F+6#\"\"#F)F)*&,&F(F )F2F/F)&F+6#\"\"$F)F)%(~.~.~.~GF)*&F2F)&F+6#,&F(F)F)F/F)F)&F+6#F(F)F)F (F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence the " }{TEXT 261 10 "Cesaro sum" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S[C];" "6#&%\"SG6#%\"C G" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" "6# -%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 9 " is the " } {TEXT 261 21 "limit of the sequence" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[n]" "6#&%\"MG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/n;" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum((n-k+1)*a[k],k = 1 .. n) = Sum((n-k+1)*a[k]/n,k \+ = 1 .. n);" "6#/-%$SumG6$*&,(%\"nG\"\"\"%\"kG!\"\"F*F*F*&%\"aG6#F+F*/F +;F*F)-F%6$*(,(F)F*F+F,F*F*F*&F.6#F+F*F)F,/F+;F*F)" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[C] = Limit(Sum((n-k+1)*a[k]/n,k = 1 .. n),n=in finity)" "6#/&%\"SG6#%\"CG-%&LimitG6$-%$SumG6$*(,(%\"nG\"\"\"%\"kG!\" \"F1F1F1&%\"aG6#F2F1F0F3/F2;F1F0/F0%)infinityG" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 22 "______________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum((-1)^n,n=1..infinity) " "6#-%$SumG6$),$\"\"\"!\"\"%\"nG/F*;F(%)infinityG" }{TEXT -1 65 " doe s not converge in the ordinary sense, but its Cesaro sum is " } {XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Sum((-1)^k,k=1..n);\ns := simplify(value(%));\nSum(s,n=1..m)/m;\n avg := simplify(value(%));\nLimit(avg,m=infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$)!\"\"%\"kG/F(;\"\"\"%\"nG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&)!\"\"%\"nG#\"\"\"\"\"##F'F+F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$,&)!\"\"%\"nG#\"\"\"\" \"##F)F-F,/F*;F,%\"mGF,F1F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$avgG ,$*&,()!\"\"%\"mGF)F*\"\"#\"\"\"F,F,F*F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,()!\"\"%\"mGF*F+\"\"#\"\"\"F-F-F+F*#F* \"\"%/F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " . . or, using the fact that the Cesaro sum of " }{XPPEDIT 18 0 "Sum(a[n],n = \+ 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" } {TEXT -1 31 " is the limit of the sequence " }{XPPEDIT 18 0 "Sum((n-k +1)*a[k]/n,k = 1 .. n);" "6#-%$SumG6$*(,(%\"nG\"\"\"%\"kG!\"\"F)F)F)&% \"aG6#F*F)F(F+/F*;F)F(" }{TEXT -1 6 " . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "a := k -> (-1)^k; \nLimit(Sum((n-k+1)/n*a(k),k=1..n),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"kG6\"6$%)operatorG%&arrowGF()! \"\"9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$SumG6$*& *&,(%\"nG\"\"\"%\"kG!\"\"F-F-F-)F/F.F-F-F,F//F.;F-F,/F,%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The last example can be generalized." }}{PARA 0 "" 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(cos*k*theta,n = 1 .. inf inity);" "6#-%$SumG6$*(%$cosG\"\"\"%\"kGF(%&thetaGF(/%\"nG;F(%)infinit yG" }{TEXT -1 65 " does not converge in the ordinary sense, but its Ce saro sum is " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "Sum(cos(k*theta),k=1..n);\ns := simplify(value( %));\nSum(s,n=1..m)/m;\navg := simplify(value(%));\nLimit(avg,m=infini ty);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$ -%$cosG6#*&%\"kG\"\"\"%&thetaGF+/F*;F+%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,$*&,,*&-%$sinG6#%&thetaG\"\"\"-F*6#*&,&%\"nGF-F- F-F-F,F-F-F-*&-%$cosGF/F--F5F+F-F-F4!\"\"F7F-F6F-F-,&F6F-F7F-F7#F7\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$,$*&,,*&-%$sinG6#%&the taG\"\"\"-F,6#*&,&%\"nGF/F/F/F/F.F/F/F/*&-%$cosGF1F/-F7F-F/F/F6!\"\"F9 F/F8F/F/,&F8F/F9F/F9#F9\"\"#/F4;F/%\"mGF/F?F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$avgG,$*&,**&%\"mG\"\"\"-%$cosG6#%&thetaGF*F*F)!\"\"- F,6#*&F.F*,&F)F*F*F*F*F/F+F*F**&F)F*,&F+F*F/F*F*F/#F/\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,**&%\"mG\"\"\"-%$cosG6#%&the taGF+F+F*!\"\"-F-6#*&F/F+,&F*F+F+F+F+F0F,F+F+*&F*F+,&F,F+F0F+F+F0#F0\" \"#/F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 " . . or, using the fact that the Cesaro sum of " }{XPPEDIT 18 0 "Sum(a[n],n = \+ 1 .. infinity)" "6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" } {TEXT -1 31 " is the limit of the sequence " }{XPPEDIT 18 0 "Sum((n-k +1)*a[k]/n,k = 1 .. n);" "6#-%$SumG6$*(,(%\"nG\"\"\"%\"kG!\"\"F)F)F)&% \"aG6#F*F)F(F+/F*;F)F(" }{TEXT -1 4 " . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "a := k -> cos(k*theta );\nLimit(Sum((n-k+1)*a(k)/n,k=1..n),n=infinity);\nsimplify(value(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"kG6\"6$%)operatorG%&a rrowGF(-%$cosG6#*&9$\"\"\"%&thetaGF1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$-%$SumG6$*&*&,(%\"nG\"\"\"%\"kG!\"\"F-F-F--% $cosG6#*&F.F-%&thetaGF-F-F-F,F//F.;F-F,/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 11 "The series " }{XPPEDIT 18 0 "Sum(sin *k*theta,n = 1 .. infinity);" "6#-%$SumG6$*(%$sinG\"\"\"%\"kGF(%&theta GF(/%\"nG;F(%)infinityG" }{TEXT -1 65 " does not converge in the ordin ary sense, but its Cesaro sum is " }{XPPEDIT 18 0 "sin*theta/(2*(1-co s*theta));" "6#*(%$sinG\"\"\"%&thetaGF%*&\"\"#F%,&F%F%*&%$cosGF%F&F%! \"\"F%F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "Sum(sin(k*theta),k=1..n);\ns := si mplify(value(%));\nSum(s,n=1..m)/m;\navg := simplify(value(%));\nLimit (avg,m=infinity);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-%$sinG6#*&%\"kG\"\"\"%&thetaGF+/F*;F+%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,$*&,**&-%$sinG6#*&,&%\"nG\"\"\"F/F/F/ %&thetaGF/F/-%$cosG6#F0F/!\"\"F)F/*&-F*F3F/-F2F+F/F/F6F4F/,&F1F/F4F/F4 #F/\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$SumG6$,$*&,**&-%$sinG 6#*&,&%\"nG\"\"\"F1F1F1%&thetaGF1F1-%$cosG6#F2F1!\"\"F+F1*&-F,F5F1-F4F -F1F1F8F6F1,&F3F1F6F1F6#F1\"\"#/F0;F1%\"mGF1F?F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$avgG,$*&,(*&-%$sinG6#%&thetaG\"\"\"%\"mGF-F-F)F--F*6 #*&F,F-,&F.F-F-F-F-!\"\"F-*&,&-%$cosGF+F-F3F-F-F.F-F3#F3\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$*&,(*&-%$sinG6#%&thetaG\" \"\"%\"mGF.F.F*F.-F+6#*&F-F.,&F/F.F.F.F.!\"\"F.*&,&-%$cosGF,F.F4F.F.F/ F.F4#F4\"\"#/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$s inG6#%&thetaG\"\"\",&-%$cosGF'F)!\"\"F)F-#F-\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " . . or, using the fact t hat the Cesaro sum of " }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity)" " 6#-%$SumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 32 " is the \+ limit of the sequence " }{XPPEDIT 18 0 "Sum((n-k+1)*a[k]/n,k = 1 .. n );" "6#-%$SumG6$*(,(%\"nG\"\"\"%\"kG!\"\"F)F)F)&%\"aG6#F*F)F(F+/F*;F)F (" }{TEXT -1 4 " . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "a := k -> sin(k*theta);\nLimit(Sum((n-k+1 )*a(k)/n,k=1..n),n=infinity);\nsimplify(value(%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"aGf*6#%\"kG6\"6$%)operatorG%&arrowGF(-%$sinG6#*&9 $\"\"\"%&thetaGF1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$ -%$SumG6$*&*&,(%\"nG\"\"\"%\"kG!\"\"F-F-F--%$sinG6#*&F.F-%&thetaGF-F-F -F,F//F.;F-F,/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$ sinG6#%&thetaG\"\"\",&-%$cosGF'F)!\"\"F)F-#F-\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "Cesaro summation of Fourier series " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 36 " be a periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 27 " which has a Fourier series" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c +``" "6#,&%\"cG\"\"\"%!GF%" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. i nfinity);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,% \"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F ,%)infinityG" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1 /(2*L)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$IntG6$-%$phiG6#%\"xG/F) ;,$%\"LG!\"\"F-" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a[k] = 1/L" "6#/&% \"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p hi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"- %$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 10 " \+ and " }{XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6#%\"kG*&\"\"\"F)%\"LG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*sin(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F* F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[0](x) = c" "6#/-&%\"sG6#\"\"!6#%\"xG%\"cG" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " s[m](x) = c+``;" "6#/-&%\"sG6#%\"mG6#%\"xG,&%\"cG\"\"\"%!GF-" } {XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. m); " "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"L G!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%\"mG" } {TEXT -1 6 ", for " }{XPPEDIT 18 0 "m >= 1" "6#1\"\"\"%\"mG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Consider the " }{TEXT 261 12 " Cesaro mean " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "sigma[n](x) = 1/(n+1)" "6#/-&%&sigmaG6#%\"nG6#%\"xG*&\" \"\"F,,&F(F,F,F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(s[m](x),m=0 ..n)" "6#-%$SumG6$-&%\"sG6#%\"mG6#%\"xG/F*;\"\"!%\"nG" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= c +``" "6#/%! G,&%\"cG\"\"\"F$F'" }{XPPEDIT 18 0 "Sum(``(1-k/(n+1))*a[k]*cos(k*Pi*x/ L)+``(1-k/(n+1))*b[k]*sin(k*Pi*x/L),k = 1 .. n);" "6#-%$SumG6$,&*(-%!G 6#,&\"\"\"F,*&%\"kGF,,&%\"nGF,F,F,!\"\"F1F,&%\"aG6#F.F,-%$cosG6#**F.F, %#PiGF,%\"xGF,%\"LGF1F,F,*(-F)6#,&F,F,*&F.F,,&F0F,F,F,F1F1F,&%\"bG6#F. F,-%$sinG6#**F.F,F9F,F:F,F;F1F,F,/F.;F,F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 261 10 "Cesaro sum" } {TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L imit(sigma[n](x),n=infinity)" "6#-%&LimitG6$-&%&sigmaG6#%\"nG6#%\"xG/F *%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``(c+Sum(``(1-k/ (n+1))*a[k]*cos(k*Pi*x/L)+``(1-k/(n+1))*b[k]*sin(k*Pi*x/L),k = 1 .. n) ),n = infinity);" "6#/%!G-%&LimitG6$-F$6#,&%\"cG\"\"\"-%$SumG6$,&*(-F$ 6#,&F,F,*&%\"kGF,,&%\"nGF,F,F,!\"\"F9F,&%\"aG6#F6F,-%$cosG6#**F6F,%#Pi GF,%\"xGF,%\"LGF9F,F,*(-F$6#,&F,F,*&F6F,,&F8F,F,F,F9F9F,&%\"bG6#F6F,-% $sinG6#**F6F,FAF,FBF,FCF9F,F,/F6;F,F8F,/F8%)infinityG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Limit(``(c+Sum(A[k,n]*cos(k*Pi*x/L)+B[k,n]*sin(k*P i*x/L),k = 1 .. n)),n = infinity);" "6#/%!G-%&LimitG6$-F$6#,&%\"cG\"\" \"-%$SumG6$,&*&&%\"AG6$%\"kG%\"nGF,-%$cosG6#**F5F,%#PiGF,%\"xGF,%\"LG! \"\"F,F,*&&%\"BG6$F5F6F,-%$sinG6#**F5F,F;F,FF,F,/F5;F,F6F,/F6%) infinityG" }{TEXT -1 16 " ------- (ii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 276 33 "_________________________________" }{TEXT -1 13 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A[k,n] = ``(1-k/(n+1))*a[k];" "6#/ &%\"AG6$%\"kG%\"nG*&-%!G6#,&\"\"\"F.*&F'F.,&F(F.F.F.!\"\"F1F.&%\"aG6#F 'F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[k,n] = ``(1-k/(n+1))*b[k]; " "6#/&%\"BG6$%\"kG%\"nG*&-%!G6#,&\"\"\"F.*&F'F.,&F(F.F.F.!\"\"F1F.&% \"bG6#F'F." }{TEXT -1 11 ", for each " }{TEXT 271 1 "k" }{TEXT -1 5 " \+ and " }{TEXT 274 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "If the Fourier series (i) converges a t " }{TEXT 272 1 "x" }{TEXT -1 54 ", then the sequence of Cesaro means also converges at " }{TEXT 275 1 "x" }{TEXT -1 63 " to a limit (Cesar o sum) (ii) which coincides with the sum (i)." }}{PARA 0 "" 0 "" {TEXT -1 44 "However, in general, there may be values of " }{TEXT 273 1 "x" }{TEXT -1 81 " for which the Cesaro sum (ii) converges, while th e series (i) does not converge." }}{PARA 0 "" 0 "" {TEXT -1 99 "In any case, the convergence properties of the two trigonometric series will be somewhat different." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Examples of Cesaro summation of Fourier series " }} {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 0 "" 0 "" {TEXT -1 47 "Consider the square wave given by th e function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " whe re " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWI SE([-1, -Pi <= x and x < 0],[1, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"x G-%*PIECEWISEG6$7$,$\"\"\"!\"\"31,$%#PiGF.F'2F'\"\"!7$F-31F4F'2F'F2" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"% #PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier se ries of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[k]*sin*k*x,k = 1 .. infinity) = Limit(``(Sum(b[k]*sin*k*x,k = 1 .. n)),n=infinity)" "6#/-%$SumG6$**&%\"bG6#%\"kG\"\"\"%$sinGF,F+F,%\"xGF,/F+;F,%)infinityG -%&LimitG6$-%!G6#-F%6$**&F)6#F+F,F-F,F+F,F.F,/F+;F,%\"nG/F?F1" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "b[k]=2*(1-(-1)^k)/(Pi*(2*k+1))" "6#/&%\"bG6#%\"kG*(\"\" #\"\"\",&F*F*),$F*!\"\"F'F.F**&%#PiGF*,&*&F)F*F'F*F*F*F*F*F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The Cesaro sum of this Fourier series is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(Sum((1-k/(n+1))*b[k]*sin*k*x,k = 1 .. n)),n = infinity);" "6#-%&LimitG6$-%!G6#-%$SumG6$*,,&\"\"\"F.* &%\"kGF.,&%\"nGF.F.F.!\"\"F3F.&%\"bG6#F0F.%$sinGF.F0F.%\"xGF./F0;F.F2/ F2%)infinityG" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The following graph compares fi nite sums " }{XPPEDIT 18 0 "Sum(b[k]*sin*k*x,k = 1 .. n)" "6#-%$SumG6$ **&%\"bG6#%\"kG\"\"\"%$sinGF+F*F+%\"xGF+/F*;F+%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Sum((1-k/(n+1))*b[k]*sin*k*x,k = 1 .. n)" "6#-%$Sum G6$*,,&\"\"\"F(*&%\"kGF(,&%\"nGF(F(F(!\"\"F-F(&%\"bG6#F*F(%$sinGF(F*F( %\"xGF(/F*;F(F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "f := x -> (-1)^floor(x/Pi): \ns := (n,x) -> Sum(2/(Pi*k)*(1-(-1)^k)*sin(k*x),k=1..n);\nsigma := (n ,x) -> \n Sum((1-k/(n+1))*(1-(-1)^k)*2/(Pi*k)*sin(k*x),k=1..n);\nplo t([f(x),s(8,x),sigma(8,x)],x=-1..10,color=[black,red,blue],\n linest yle=[2,1,1],numpoints=100,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$SumG6$,$*&*&,& \"\"\"F4)!\"\"%\"kGF6F4-%$sinG6#*&F7F49%F4F4F4*&%#PiGF4F7F4F6\"\"#/F7; F49$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6$%\"nG%\"xG6 \"6$%)operatorG%&arrowGF)-%$SumG6$,$*&*(,&\"\"\"F4*&%\"kGF4,&9$F4F4F4! \"\"F9F4,&F4F4)F9F6F9F4-%$sinG6#*&F6F49%F4F4F4*&%#PiGF4F6F4F9\"\"#/F6; F4F8F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 389 178 178 {PLOTDATA 2 "6(-% 'CURVESG6%7\\t7$$!\"\"\"\"!F(7$$!16666^[P))!#;F(7$$!1AAAAr)f#yF.F(7$$! 1XWWW=X)o'F.F(7$$!1XWWWiPVbF.F(7$$!1*)))))))Hu.WF.F(7$$!1yxxxq:ZLF.F(7 $$!1nmmmN7`AF.F(7$$!1cbbb!)R(o\"F.F(7$$!1XWWWDn@6F.F(7$$!1VLLL$prR)!#< F(7$$!1LAAAKhxbFJF(7$$!1ymmm^$y;%FJF(7$$!1B666r0eFFJF(7$$!1XLL$3oJ0#FJ F(7$$!1obbb!z#[8FJF(7$$!1#zmmTX$e**!#=F(7$$!1/zxx-!RV'FgnF(7$$!1$\"\"\"F*7$$\"1Xbbb!>?'eFJFbo7$$\"1566 6RDm6F.Fbo7$$\"1bbbbT2)=#F.Fbo7$$\"1MLLLyF.Fbo7$$\"1VW WW!=v$))F.Fbo7$$\"1+++IM>-5!#:Fbo7$$\"1xxxPV.26FbqFbo7$$\"1+++I91A7Fbq Fbo7$$\"1LLLBWfJ8FbqFbo7$$\"166661)eW\"FbqFbo7$$\"1yxx25$3b\"FbqFbo7$$ \"1*))))))=MSm\"FbqFbo7$$\"1*)))))y/i\"y\"FbqFbo7$$\"1+++!pzR)=FbqFbo7 $$\"1AAAU/`%*>FbqFbo7$$\"1mmm'=S(3@FbqFbo7$$\"1+++S@Z?AFbqFbo7$$\"1+++ ]&y&GBFbqFbo7$$\"1mmm'G7'[CFbqFbo7$$\"1xxxPzYcDFbqFbo7$$\"1nmmmoirEFbq Fbo7$$\"1AAAsz(fx#FbqFbo7$$\"1LLL$Hh+*GFbqFbo7$$\"1xxx2[S(*HFbqFbo7$$ \"1*)))))es]`IFbqFbo7$$\"1+++5(4'4JFbqFbo7$$\"1*))Q,JAL7$FbqFbo7$$\"1x xF5\\.PJFbqFbo7$$\"1*\\7.1j/9$FbqFbo7$$\"1AsM57*Q9$FbqF(7$$\"1W>Qg$>t9 $FbqF(7$$\"1mmT5vu]JFbqF(7$$\"16h[5QgdJFbqF(7$$\"1bbb5,YkJFbqF(7$$\"1L L$3J&)=>$FbqF(7$$\"166660J>KFbqF(7$$\"1AAA(zLnF$FbqF(7$$\"1KLL$3dTL$Fb qF(7$$\"1AAA#)zwWMFbqF(7$$\"1AAA-O)yb$FbqF(7$$\"1xxxnD1qOFbqF(7$$\"1nm m'QVJx$FbqF(7$$\"1yxxPcG\"*QFbqF(7$$\"1bbbvM&p*RFbqF(7$$\"1+++Iuh4TFbq F(7$$\"1+++57X!em%FbqF(7$$\"1nmmc&>Jy%FbqF(7$$\"1@AAU[M%)[Fbq F(7$$\"1nmmE'G'**\\FbqF(7$$\"1666rq)46&FbqF(7$$\"1yxxnFFA_FbqF(7$$\"1W WWk!\\JL&FbqF(7$$\"1WWWukmRaFbqF(7$$\"1bbb:Y\"[b&FbqF(7$$\"1LLLV\"fXm& FbqF(7$$\"1566\"4g+y&FbqF(7$$\"1AAAiih%)eFbqF(7$$\"1WWW//<+gFbqF(7$$\" 1nmmO6&36'FbqF(7$$\"1yxx-o0mhFbqF(7$$\"1))))))oCE@iFbqF(7$$\"1*****\\t $4]iFbqF(7$$\"1566,]#*yiFbqF(7$$\"1**\\P4*GDG'FbqF(7$$\"1)))Qw\"G8'G'F bqFbo7$$\"1xF!fsO(*G'FbqFbo7$$\"1mm;M1M$H'FbqFbo7$$\"1WWp]%[0I'FbqFbo7 $$\"1AAAniv2jFbqFbo7$$\"1zxF+>sFbqFbo7$$\"1+++!eI&HtFbqFbo7$$\"1!*))))) ytrW(FbqFbo7$$\"1WWWC)))4b(FbqFbo7$$\"1VWWC)p2n(FbqFbo7$$\"1AAAUH')yxF bqFbo7$$\"1AAAKa\"e)yFbqFbo7$$\"1xxxm)FbqFbo7$$\"1+++!H\"Gz()FbqFbo7$$ \"1AAA#*)\\&*)))FbqFbo7$$\"1+++?Mv)**)FbqFbo7$$\"1******>Q/3\"*FbqFbo7 $$\"1******4J^>#*FbqFbo7$$\"1********3fy#*FbqFbo7$$\"1*******oowL*FbqF bo7$$\"1mm;kHvk$*FbqFbo7$$\"1KLLQs$=R*FbqFbo7$$\"1lmTv$z`S*FbqFbo7$$\" 1****\\7:#*=%*FbqFbo7$$\"1L3xYqIA%*FbqFbo7$$\"1m;/\"e#pD%*FbqF(7$$\"1+ DJ:\"y!H%*FbqF(7$$\"1LLe\\OYK%*FbqF(7$$\"1+]7=ZBR%*FbqF(7$$\"1mmm'y0gW *FbqF(7$$\"1WWW*4x%*\\*FbqF(7$$\"1@AA7%[Hb*FbqF(7$$\"1*)))))=$osm*FbqF (7$$\"1566h$R8y*FbqF(7$$\"1cbb&*[s$))*FbqF(7$$\"#5F*F(-%'COLOURG6&%$RG BGF*F*F*-%*LINESTYLEG6#\"\"#-F$6%7e\\l7$F($!1_C4_%*f15Fbq7$$!1bbbbDu=% *F.$!1/O%Q,:Li*F.7$F,$!1,B61X&)G#*F.7$$!1)))))Qh5Ye)F.$!1:o6_Q*e4*F.7$ $!1mmm;htJ$)F.$!1;')y!)F.$!1=?$*GiJN*)F.7$F0$!1N# )f&\\,x\"*)F.7$$!1xxx-LgTvF.$!1S]6$3JJ&*)F.7$$!1MLL$[>sD(F.$!1r/fsIf[! 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3,!*F.7$F_bl$\"1lX!*o?7X*)F.7$Fbgp$\"1+z%>!y\"3#*)F.7$Fbbl$\"1S=It\"f$ >*)F.7$F_hp$\"1Y+p!Rhw$*)F.7$Febl$\"1o/cIvFd*)F.7$$\"1bbbS%[o!))Fbq$\" 17\"Qz(3?e*)F.7$$\"1566\"f:W$))Fbq$\"1B/xJ7N[*)F.7$$\"1lmmTF)>'))Fbq$ \"1]Y)\\EiK#*)F.7$Fhbl$\"1\"y\"34<@y))F.7$Fjhp$\"1b(=//S5r)F.7$F[cl$\" 1-RZ*)yg4%)F.7$Fgip$\"13FM;S%4%zF.7$F^cl$\"1W`hBbB\"G(F.7$Fdjp$\"1G0nX qT*R'F.7$Facl$\"1,L$4NO)4`F.7$Fdcl$\"1pDpGMu]RF.7$Fgcl$\"1+LH(pCYU#F.7 $F]dl$\"1>t\"48T%)H*FJ7$Feel$!1UewyeQ+gFJ7$Fhel$!1h9[fK@)3#F.7$F[fl$!1 *oBo!\\`+NF.7$Fh]q$!1iwyB3,t[F.7$F^fl$!14$p[us*fgF.7$F`^q$!1kkt!Qj;.(F .7$Fafl$!1*p-:yr;y(F.7$F]_q$!1F3SN[!HF)F.7$Fdfl$!1gDSyjQ4')F.7$$\"1mmm rOz7**Fbq$!1w-pm!G2u)F.7$Fj_q$!1(GP%=HKN))F.7$$\"1!*)))QAJ4(**Fbq$!1) \\h#3Fw)*))F.7$Fgfl$!15SSYdvO*)F.-Fjfl6&F\\glF*F*Fc`qFf`q-%+AXESLABELS G6$Q\"x6\"%!G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F(FgflFeis" 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" }}}}{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 0 "" 0 "" {TEXT -1 53 "Consider again the square wave given by the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([-1, -Pi < = x and x < 0],[1, 0 <= x and x < Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISE G6$7$,$\"\"\"!\"\"31,$%#PiGF.F'2F'\"\"!7$F-31F4F'2F'F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " an d " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We obtain better results by applying the Cesaro summation to the \+ " }{TEXT 261 14 "non-zero terms" }{TEXT -1 55 " of the Fourier series \+ by giving the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 13 " in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(beta[k]*sin((2*k+1)*x),k = 0 .. infinity) = Limit(` `(Sum(beta[k]*sin((2*k+1)*x),k = 0 .. n)),n = infinity);" "6#/-%$SumG6 $*&&%%betaG6#%\"kG\"\"\"-%$sinG6#*&,&*&\"\"#F,F+F,F,F,F,F,%\"xGF,F,/F+ ;\"\"!%)infinityG-%&LimitG6$-%!G6#-F%6$*&&F)6#F+F,-F.6#*&,&*&F3F,F+F,F ,F,F,F,F4F,F,/F+;F7%\"nG/FKF8" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "beta[k] = 4/(Pi*(2*k+1)) ;" "6#/&%%betaG6#%\"kG*&\"\"%\"\"\"*&%#PiGF*,&*&\"\"#F*F'F*F*F*F*F*!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k=0,1,` . . . `" "6%/%\"kG\"\"! \"\"\"%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "The Cesaro sum of this Fourier series is \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(Sum((1-k /(n+1))*beta[k]*sin((2*k+1)*x),k = 1 .. n)),n = infinity);" "6#-%&Limi tG6$-%!G6#-%$SumG6$*(,&\"\"\"F.*&%\"kGF.,&%\"nGF.F.F.!\"\"F3F.&%%betaG 6#F0F.-%$sinG6#*&,&*&\"\"#F.F0F.F.F.F.F.%\"xGF.F./F0;F.F2/F2%)infinity G" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "The following graph compares finite sums \+ " }{XPPEDIT 18 0 "Sum(beta[k]*sin((2*k+1)*x),k = 0 .. n);" "6#-%$SumG6 $*&&%%betaG6#%\"kG\"\"\"-%$sinG6#*&,&*&\"\"#F+F*F+F+F+F+F+%\"xGF+F+/F* ;\"\"!%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Sum((1-k/(n+1))*beta[ k]*sin((2*k+1)*x),k = 0 .. n);" "6#-%$SumG6$*(,&\"\"\"F(*&%\"kGF(,&%\" nGF(F(F(!\"\"F-F(&%%betaG6#F*F(-%$sinG6#*&,&*&\"\"#F(F*F(F(F(F(F(%\"xG F(F(/F*;\"\"!F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "f := x -> (-1)^floor(x/Pi):\ns := (n,x) -> Sum(4/(Pi*(2*k+1))*sin ((2*k+1)*x),k=0..n);\nsigma := (n,x) -> Sum((1-k/(n+1))*4/(Pi*(2*k+1)) *sin((2*k+1)*x),k=0..n);\nplot([f(x),s(8,x),sigma(8,x)],x=-1..10,color =[black,red,blue],\n linestyle=[2,1,1],numpoints=100,ytickmarks=3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"nG%\"xG6\"6$%)operato rG%&arrowGF)-%$SumG6$,$**\"\"%\"\"\"%#PiG!\"\",&*&\"\"#F3%\"kGF3F3F3F3 F5-%$sinG6#*&F6F39%F3F3F3/F9;\"\"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$Sum G6$,$*,\"\"%\"\"\",&F3F3*&%\"kGF3,&9$F3F3F3!\"\"F9F3%#PiGF9,&*&\"\"#F3 F6F3F3F3F3F9-%$sinG6#*&F;F39%F3F3F3/F6;\"\"!F8F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 387 177 177 {PLOTDATA 2 "6(-%'CURVESG6%7\\t7$$!\"\"\"\"!F (7$$!3z5666^[P))!#=F(7$$!3:AAAAr)f#yF.F(7$$!3=XWWW=X)o'F.F(7$$!3aWWWWi 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