Taylor Series: Assume |n-th derive of f| <= 3 for all n, x

[lislr8]

Hi I'm having difficulty with this question. I'm not sure which Taylor series to evaluate it with.

Assume that f is a function where the absolute value of the nth derivative at x is always less than or equal to 3 for all n and all real x. Find the least integer n for which you can be sure that Pn (1/2) approximates f(1/2) with a four decimal place accuracy (to 0.00005).

The answer is n=5. If it said that the nth derivative was always less than or equal to 1 I would have used sin or cosine. I tried using 3 sin, but the n that I got was 7. How do you know which function to use to test it in? I was thinking about e^x since e is less than or equal to 3 but that didn't work out either.

Thanks for your help,

lislr8

 

[Unco]

Re: Taylor Series

Hi lislr,

Look in your notes (or the web) for info about the remainder of a taylor approximation to a function.

You may find something about the remainder for the nth order taylor series to a function at a point x being given by
\(\displaystyle R_n(x) = \frac{f^{n+1}(c)}{(n+1)!}x^{n+1}\)
where c is some constant in [0,x]. When talking about the error we take the absolute value.

 

[lislr8]

Re: Taylor Series

Thanks, I appreciate the help!