Summation by parts interpretation

Question

So in Rudin, I'm given the theorem for summation by parts: $$\text{Given two sequences} \{a_n\}, \{b_n\}, \text{put}\\ A_n = \sum_{k=0}^n a_k\\$$ $\text{if}$ $$n \ge 0; \text{put}\ A_{-1} = 0$$ Then, if $$0 \le p \le q$$ we have $$\sum_{n=p}^q a_n b_n = \sum_{n=p}^{q-1}A_n(b_n - b_{n+1}) + A_qb_q - A_{p-1}b_p$$ I'm having trouble understanding when/how I would use this. $\\$All it says in the book is "useful in the investigation of series of the form $\sum a_nb_n$ when $b_n$ is monotone". I understand that it is meant to be used to see if $\sum a_nb_n$ converges or not, but I don't know how to use it. $\\$Also, I'm confused by the subscripts; what is the $A_{-1}$ and where does this come into play?

Answer 1

The most typical uses are in the proof of the fact that if $\sum a_n$ converges and $b_n$ is bounded and monotone, then $\sum a_nb_n$ converges. You can see a proof in the Wikipedia article.

Another very classical application is Abel's Theorem. You can see summation by parts used several times in these notes.

As for $A_{-1}$, it allows you to write the last formula for all $p\geq0$. Otherwise you would have to distinguish the case $p=0$.