# Summation by parts interpretation

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?

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.
As for $$A_{-1}$$, it allows you to write the last formula for all $$p\geq0$$. Otherwise you would have to distinguish the case $$p=0$$.
• Well, "only" is a very strong word; I have probably read about $1/10^6$ of all the math ever written, so I cannot guarantee anything. But yes, you mostly use it as a trick to manipulate a series of that form. It has a name simply because it is the discrete analog of integration by parts. Commented Nov 15, 2015 at 1:16