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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?

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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 $p=0$.

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  • $\begingroup$ So is it really only useful as a base for proving other theorems? $\endgroup$
    – user181928
    Commented Nov 15, 2015 at 1:10
  • $\begingroup$ 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. $\endgroup$ Commented Nov 15, 2015 at 1:16
  • $\begingroup$ Now, after googling "summation by parts", here an example. $\endgroup$ Commented Nov 15, 2015 at 1:16
  • $\begingroup$ Ha ok that's fair, thanks for your example though! And yeah, I got the connection to integration by parts. I just thought since that has a lot of practical implications, that this might as well.. $\endgroup$
    – user181928
    Commented Nov 15, 2015 at 2:43

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