1. ## taylor series problem

Given a function f such that f(3)= 1 and $\displaystyle f^{(n)}(3)=\frac{(-1)^n n!}{(2n+1)2^n}$

(a) write the first four nonzero terms and the general term of the Taylor series for f around x= 3

(b) find the radius of convergence of the Taylor series

(c) show that the third degree Taylor polynomial approximates f(4) within 0.01

2. Originally Posted by yoman360
Given a function f such that f(3)= 1 and $\displaystyle f^{(n)}(3)=\frac{(-1)^n n!}{(2n+1)2^n}$

(a) write the first four nonzero terms and the general term of the Taylor series for f around x= 3

(b) find the radius of convergence of the Taylor series

(c) show that the third degree Taylor polynomial approximates f(4) within 0.01
(a) You should know that

$\displaystyle f(x) = \sum_{n = 0}^{\infty}\frac{f^{(n)}(a)}{n!}(x - a)^n$.

In this case, $\displaystyle a = 3$.

Substitute all your known information, then write the first four nonzero terms.

(b) Find $\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right|$.

The series converges when this limit is less than 1.

So solve $\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| < 1$ for $\displaystyle x$.