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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# CESARO SUMMATION O
F A TRIANGLE WAVE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "aodd := k -> 1/(2*k + 1)^2;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "partialsum := K -> (Pi/
2) - (4/Pi)*sum(aodd(k)*cos((2*k+1)*x), k=0..K);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 50 "mean := M -> (1/(M+1))*sum(partialsum(K), K=0
..M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([partialsum(1
), mean(1)], x=-1..7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p
lot([partialsum(5), mean(5)], x=-1..7);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 42 "plot([partialsum(20), mean(20)], x=-1..7);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "# Close inspection shows tha
t the Cesaro mean curve is BLUNTER than the partial sum curve." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "# When the partial sums are
already well convergent, Cesaro summation just slows down the converg
ence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# This is shown cl
early by examining the error in the approximations:" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 57 "plot([abs(x) - partialsum(5), abs(x) - mean(5)], x=0..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([abs(x) - partialsum(10
), abs(x) - mean(10)], x=0..1);" }}}}{MARK "14" 0 }{VIEWOPTS 1 1 0 1
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }