Question about Rearrangement of Absolutely convergent Series

Question

Suppose that $\sum_{n=1}^\infty a_n$ is an absolutely convergent series and let $f : \mathbb{N} \to \mathbb{N}$ be a bijective map. Then the series $\sum_{n=1}^\infty a_{f(n)}$ is absolutely convergent and has the same limit $L$.

I was reading this and I was a little unclear as to what this was saying. Is this statement saying that $\sum_{n=1}^\infty a_{f(n)} = L$, or is it saying that $\sum_{n=1}^\infty |a_{f(n)}| = L$? Because I can prove the former and I can show that the series $\sum_{n=1}^\infty |a_{f(n)}|$ converges but I don't know whether this statement is saying that $\sum_{n=1}^\infty |a_{f(n)}| = L$ as well.

Answer 1

This says that it is given that $\sum_{n=1}^\infty a_n = L < \infty$ and $\sum_{n=1}^\infty |a_n| = M < \infty$. Then $\sum_{n=1}^\infty a_{f(n)} = L$ and $\sum_{n=1}^\infty |a_{f(n)}| = M$.

In short, arbitrary rearrangement of the terms of an absolutely convergent series does not alter the convergence of the series. (This is automatic for (absolutely) convergent series having only nonnegative terms, such as the two series having value $M$ above.)

Answer 2

It is saying the following. Assume that $\sum_{n=1}^{+ \infty} a_n$ converges absolutely, and that $$ \sum_{n=1}^{+ \infty} a_n = L. $$ Then $\sum_{n=1}^{+ \infty} a_{f(n)}$ converges absolutely, and $$ \sum_{n=1}^{+ \infty} a_{f(n)} = L. $$ There are therefore two things to prove, but the proofs are essentially the same.

Answer 3

It's saying that if $\sum_{n=1}^\infty |a_n|<+\infty$ and $\sum_{n=1}^\infty a_n = L$ then $\sum_{n=1}^\infty a_{f(n)}=L.$