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In the same vein as this question

Striking applications of integration by parts

I'd also like to have a list of some good applications of the discrete version: summation by parts.

Community
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user908123
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    if $e^{ix} \ne 1$ : $$\sum_{n=1}^N \frac{e^{i n x}}{n} = \frac{1}{N}\sum_{n=1}^N e^{inx} + \sum_{n=1}^{N-1} (\sum_{k=1}^n e^{ikx}) (\frac{1}{n}-\frac{1}{n+1}) = \frac{1-e^{i (N+1) x}}{N (1-e^{ix})} +\sum_{n=1}^{N-1} \frac{1-e^{i (n+1) x}}{n (n+1) (1-e^{ix})}$$ hence $\sum_{n=1}^\infty \frac{e^{i n x}}{n}$ converges. In the same way you can show that $\sum_{n=1}^\infty \frac{e^{i n x}}{n^a}$ converges whenever $a > 0$ – reuns Jun 19 '16 at 00:37

2 Answers2

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Summation by parts is useful when the summand has a harmonic number $H_n$ in it. Recall that summation by parts states

$$\sum_{n=1}^m f_n (g_{n+1}-g_n) = f_{m+1} g_{m+1} - f_1 g_1 - \sum_{n=1}^m g_{n+1} (f_{n+1}-f_n) $$

Note that a Harmonic number difference is very simple (e.g., $\frac1{n+1}$).

For example, consider

$$\sum_{n=1}^{\infty} \frac{H_n}{n (n+1)} $$

Note that $g_n = -\frac1n$. Then the sum is

$$0 - \left ( -\frac{H_1}{1}\right ) - \sum_{n=1}^{\infty} \left [-\frac1{(n+1)^2}\right ] = 1 + \frac{\pi^2}{6} - 1 $$

Thus the sum is

$$\sum_{n=1}^{\infty} \frac{H_n}{n (n+1)} = \frac{\pi^2}{6}$$

which is not trivially derived otherwise.

ADDENDUM

The above result leads to the very interesting-looking summation:

$$\sum_{n=1}^{\infty} \frac{1+\frac12+\frac13+\cdots+\frac1n}{1+2+3+\cdots+n} = \frac{\pi^2}{3} $$

Ron Gordon
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  • Thank you for providing this example. More generally when we want to establish properties about $\sum a_n b_n$ summation by part is good tool to use. Nevertheless it is not always easy (at least for me) to see which one is going to be the "discrete derivative" and which one is going to be the "discrete antiderivative". In the example your provided $H_n$ is going to be the discrete derivative for example, it's not to difficult to see since $H_{n+1}-H_n$ is very nice and slow. – Thinking Nov 12 '18 at 10:48
  • But more generally what could be the intution of taking $a_n$ as the discrete derivative and $b_n$ the discrete antiderivative or the reverse ? Thanl you ! – Thinking Nov 12 '18 at 10:49
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Summation by parts is used to prove Abel's Theorem, and is also used in the Cauchy Criterion when testing for convergence.
Additionally, the following paper given here proves to be rather interesting.

Summation by parts also find many application in recreational mathematics such as IMO questions.

Anthony Peter
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