Chapter 6.1, Problem 57E

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.

57. [T] Plot the graphs of $\frac{1}{1-x}$ and of the partial sums $S_N={\displaystyle \underset{n=0}{\overset{N}{\sum }}{x}^n}$ for $n=10,20,30$ on the interval [-0.99, 0.99]. Comment on the approximation of $\frac{1}{1-x}$ by $S_N$ near $x=-1$ and near $x=1$ as N increases.