Section 1
Power series
Complete the proof of Theorem 6.1 .1 by showing that series (4) diverges when $|x|>r$.
Prove: If the radius of convergence of $\sum_{n=0}^{\infty} a_n x^n$ is $r$, then the radius of convergence of $\sum_{n=0}^{\infty} a_n x^{k n}$ is $r^{1 / k}$ for every $k \in \mathbb{N}$.
If the radius of convergence of $\sum_{n=0}^{\infty} a_n x^n$ is $r$ and $\lim _{n \rightarrow \infty} b_n=1$, what can you say about the radius of convergence of $\sum_{n=0}^{\infty} a_n b_n x^n$.
If the radius of convergence of $\sum_{n=0}^{\infty} a_n x^n$ is $r$ and the sequence ( $b_n$ ) is convergent, what can you say about the radius of convergence of $\sum_{n=0}^{\infty} a_n b_n x^n$.
If the radius of convergence of $\sum_{n=0}^{\infty} a_n x^n$ is $r$ and the sequence $\left(b_n\right)$ is bounded, what can you say about the radius of convergence of $\sum_{n=0}^{\infty} a_n b_n x^n$.
Find the radius of convergence of the series $\sum_{n=0}^{\infty} n x^n$. (Hint: use Theorem 5.6.5.)