How do you find the smallest value of #n# for which the Taylor Polynomial #p_n(x,c)# to approximate a function #y=f(x)# to within a given error on a given interval #(c-r,c+r)#?

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Answer 1 · 2014-09-20T13:56:11

Let #epsilon>0# be an acceptable error.

The error can be estimated by

#|f(x)-p_n(x,c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|#,
where #z# in #(c-r,c+r)#.

since #x# in #(c-r,c+r)#,

#\leq |f^{(n+1)}(z)|/{(n+1)!}r^{n+1}#

if we know that the absolute value of all derivatives of $f(x)$ are bounded by some fix value #M#, then

#leq M/{(n+1)!}r^{n+1}#

Find #n# such that

#M/{(n+1)!}r^{n+1} le epsilon#