Chapter 6.1, Problem 61E

In the following exercises, suppose that $p(x)=$ ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n{x}^n}$ Satisfies $\underset{n\to \infty }{\lim }$

$\frac{a_n+1}{a_n}=1$ where $a_n\ge 0$ for each $n$ . State whether each series converges on the full interval (− 1, 1), or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.

61. [T] Plot the graphs of the partial sums $S_N={\displaystyle \underset{n=0}{\overset{N}{\sum }}{(-1)}^n\frac{x^{2n}+1}{(2n+1)!}}$ For n =3, 5, 10 on the interval $[-2\pi ,2\pi ]$ . Comment on the how these plots approximate $\sin x$ as N increases.