Chapter 6.1, Problem 55E

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.

55. ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_{n}2x^{n^2}}$

$(H\mathrm{int}:\text{ Let }{b}_k=a_k\text{ if }k=n^2$ for

some $n$ otherwise $b_k=0)$ .