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for the function 1/x a fourth degree taylor series approximation cente
red at c = 1 is " }}{PARA 0 "" 0 "" {TEXT -1 4 "1/x " }{OLE 1 3584 1 "
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92 "The 5th derivative of 1/x is -5!/x^6 so for x > 1 the maginitude o
f the fifth derivative is " }{OLE 1 3584 1 "[xm]Br=WfoRrB:::wk;nyyI;G:
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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::1:" }{TEXT 
-1 3 " 5!" }}{PARA 0 "" 0 "" {TEXT -1 49 "and thus our error, Rn(x),  \+
is bounded by (x-1)^5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 52 "If x = 1.72, the upper bound on the approximation is" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "maxError:= (1.72 -1)^5;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)maxErrorG$\"+Kw\"\\$>!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "The actual error is" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "actualError:= 1/1.72 - (1 - .72 + .72^2 - .72^3 + .72
^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,actualErrorG$!+7@&\\7\"!#5
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Thus our actual error is quit
e a bit less." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 65 "Find an plot the sum of 1 - 1/x + 1/x^2 - ... + 1/x^n for
 n = 100" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "f:= 1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Generat
e the partial sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for n from 1 t
o 50 do\nf:= f + x^(-n)*(-1)^n:\nend do:\n" }}{PARA 0 "" 0 "" {TEXT 
-1 21 "Plot the partial sum." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "plot(f,x = 1 .. 10);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f2:= sum('(-1)^n/(2*n +1)!*x^(2*n +
1)', 'n' = 0..50):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f2,x = -
50..50);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" 
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