I have this form for summation by parts from an exercise.
$$\sum_{n=M}^{N} a_n b_n = a_N \sum_{n=1}^N b_n -a_M\sum_{n=1}^{M-1}b_n - \sum_{n=M}^{N-1} \Big[ (a_{n+1} - a_{n}) \sum_{i=1}^n b_n \Big]$$
The next exercise supposes that $\sum_{n=1}^{\infty} a_n$ converges and asks me to show
$$\lim_{r\to 1^-} \sum_{n=1}^{\infty} r^n a_n = \sum_{n=1}^{\infty} a_n $$
It suggests using the summation by parts formula above, so I did. I choose $M=1$, rewrite the infinite sum as a limit, and use the convention that an empty sum is $0$.
$$\lim_{r\to 1^-} \lim_{N \to \infty} r^N \sum_{n=1}^{N} a_n - 0 - \sum_{n=1}^{N-1} \Big[ (r^{n+1} - r^{n}) \sum_{i=1}^n a_n \Big] $$
It's clear from the comparison test that if $\sum_{n=1}^{\infty} a_n$ converges, $\sum_{n=1}^{\infty} r^n a_n$ must also converge, so I feel like these limits must converge as well. Can I use that to justify switching the limits? If so the problem becomes easy as each $r$ becomes a 1 like so.
$$\lim_{N \to \infty} \lim_{r\to 1^-} r^N \sum_{n=1}^{N} a_n - \sum_{n=1}^{N-1} \Big[ (r^{n+1} - r^{n}) \sum_{i=1}^n a_n \Big] $$
$$= \lim_{N \to \infty} 1 \sum_{n=1}^{N} a_n - \sum_{n=1}^{N-1} \Big[ (1 - 1) \sum_{i=1}^n a_n \Big] $$
$$= \lim_{N \to \infty} \sum_{n=1}^N a_n $$
So, is switching the limits like this justified? When can I tell when switching limits like this is justified? If I know a limit converges, am I free to do more or less anything, other than breaking it into pieces that don't converge on their own?
This is from chapter 1 of Stein and Shakarchi's complex analysis book, and is said to be related to being "Abel Summable" but all other questions on that look like this.
