# 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)#?

One way to estimate the error is by

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To find the smallest value of ( n ) for which the Taylor Polynomial ( p_n(x,c) ) approximates a function ( y = f(x) ) within a given error on a given interval ((c-r, c+r)), you can use the remainder term formula of Taylor's theorem, which is given by:

[ R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1} ]

where ( z ) is some point between ( x ) and ( c ). You want ( R_n(x) ) to be less than or equal to the given error on the interval ((c-r, c+r)). So, you set up the inequality:

[ \left| \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1} \right| \leq \text{error} ]

Then, solve for ( n ) by finding the smallest integer that satisfies this inequality.

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