PlanetMath: example of summation by parts

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example of summation by parts

(Example)

Proposition. The series $\displaystyle\sum_{n=1}^\infty\frac{\sin{n\varphi}}{n}$ and $\displaystyle\sum_{n=1}^\infty\frac{\cos{n\varphi}}{n}$ converge for every complex value $\varphi$ which is not an even multiple of $\pi$ .

Proof. Let $\varepsilon$ be an arbitrary positive number. One uses the identities

$\displaystyle \sin{\varphi}+\sin{2\varphi}+\ldots+\sin{n\varphi} = \frac{\sin(n+\frac{1}{2})\varphi-\sin\frac{\varphi}{2}}{2\sin\frac{\varphi}{2}},$

(1)

$\displaystyle \cos{\varphi}+\cos{2\varphi}+\ldots+\cos{n\varphi} = \frac{-\cos(n+\frac{1}{2})\varphi+\cos\frac{\varphi}{2}}{2\sin\frac{\varphi}{2}},$

(2)

proved in the entry “example of telescoping sum”. These give the estimates

$\displaystyle \vert\sin{\varphi}+\sin{2\varphi}+\ldots+\sin{n\varphi}\vert \leqq \frac{2}{2\vert\sin\frac{\varphi}{2}\vert}\, :=\, K_\varphi,$

$\displaystyle \vert\cos{\varphi}+\cos{2\varphi}+\ldots+\cos{n\varphi}\vert \leqq \frac{2}{2\vert\sin\frac{\varphi}{2}\vert}\, :=\, K_\varphi$

for any $n = 1,\,2,\,3,\,\ldots$ . We want to apply to the series $\sum_{n=1}^\infty\frac{\cos{n\varphi}}{n}$ the Cauchy general convergence criterion for series. Let us use here the short notation

$\displaystyle \cos{N\varphi}+\cos{(N\!+\!1)\varphi}+\ldots+\cos{(N\!+\!p)\varphi} := S_{N,N+p}\quad (p = 0,\,1,\,2,\,\ldots).$

Then, utilizing Abel's summation by parts, we obtain

$\displaystyle \left\vert\sum_{n=N}^{N+P}\frac{\cos{n\varphi}}{n}\right\vert = \... ...c{1}{N\!+\!p\!+\!1}\right)S_{N,N+p}+\frac{1}{N\!+\!P}S_{N,N+P}\right\vert \leqq$

$\displaystyle \leqq \sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)\vert S_{N,N+P}\vert +\frac{1}{N+P}\vert S_{N,N+P}\vert <$

$\displaystyle < \sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}... ...ot2K_\varphi+\frac{1}{N\!+\!P}\cdot2K_\varphi\, =\, \frac{1}{N}\cdot2K_\varphi;$

the last form is gotten by telescoping the preceding sum and before that by using the identity

$\displaystyle S_{N,N+p} = [\cos\varphi+\cos2\varphi+\ldots+\cos(N\!+\!p)\varphi]-[\cos\varphi+\cos2\varphi+\ldots +\cos(N\!-\!1)\varphi].$

Thus we see that

$\displaystyle \left\vert\sum_{n=N}^{N+P}\frac{\cos{n\varphi}}{n}\right\vert < \frac{2K_\varphi}{N} < \varepsilon$

for all natural numbers

as soon as $N > \frac{2K_\varphi}{\varepsilon}$ . According to the Cauchy criterion, the latter series is convergent for the mentioned values of $\varphi$ . The former series is handled similarly.

"example of summation by parts" is owned by pahio.

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Cross-references: convergent, natural numbers, sum, summation by parts, number, positive, even multiple, complex, converge, series, proposition
There are 2 references to this entry.

This is version 5 of example of summation by parts, born on 2007-08-10, modified 2007-08-10.
Object id is 9849, canonical name is ExampleOfSummationByParts.
Accessed 897 times total.

Classification:

AMS MSC:

40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

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