How do you find the largest interval #(c-r,c+r)# on which the Taylor Polynomial #p_n(x,c)# approximates a function #y=f(x)# to within a given error?

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Answer 1 · 2014-10-06T23:44:53

Let us assume that there is #M>0# such that

#|f^{(n+1)}(x)| le M# for all #x#.

If we want the error to be less than #epsilon>0#, then

#|R_n(x;c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|#,

where #z# is between #x# and #c#.

by replacing #|f^{(n+1)}(z)|# by #M#,

#le M/{(n+1)!}|x-c|^{n+1} < epsilon#

By solving for #|x-c|#,

#Rightarrow |x-c| < root(n+1){{epsilon (n+1)!}/M}#

Hence,

#r=root(n+1){{epsilon (n+1)!}/M}#

I hope that this was helpful.