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example of summation by parts
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(Example)
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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
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proved in the entry ``example of telescoping sum''. These give the estimates $$|\sin{\varphi}+\sin{2\varphi}+\ldots+\sin{n\varphi}| \leqq \frac{2}{2|\sin\frac{\varphi}{2}|}\, :=\, K_\varphi,$$ $$|\cos{\varphi}+\cos{2\varphi}+\ldots+\cos{n\varphi}| \leqq \frac{2}{2|\sin\frac{\varphi}{2}|}\, :=\, K_\varphi$$ 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 $$\cos{N\varphi}+\cos{(N\!+\!1)\varphi}+\ldots+\cos{(N\!+\!p)\varphi} := S_{N,N+p}\quad (p = 0,\,1,\,2,\,\ldots).$$ Then, utilizing Abel's summation by parts, we obtain $$\left|\sum_{n=N}^{N+P}\frac{\cos{n\varphi}}{n}\right| = \left|\sum_{p=0}^{P}\frac{1}{N\!+\!p}\cos{(N+p)\varphi}\right| = \left|\sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)S_{N,N+p}+\frac{1}{N\!+\!P}S_{N,N+P}\right| \leqq$$ $$\leqq \sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)|S_{N,N+P}| +\frac{1}{N+P}|S_{N,N+P}| <$$ $$< \sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)\cdot2K_\varphi+\frac{1}{N\!+\!P}\cdot2K_\varphi\, =\, \frac{1}{N}\cdot2K_\varphi;$$ the last form is gotten by telescoping the preceding sum and before that by
using the identity $$S_{N,N+p} = [\cos\varphi+\cos2\varphi+\ldots+\cos(N\!+\!p)\varphi]-[\cos\varphi+\cos2\varphi+\ldots +\cos(N\!-\!1)\varphi].$$ Thus we see that $$\left|\sum_{n=N}^{N+P}\frac{\cos{n\varphi}}{n}\right| < \frac{2K_\varphi}{N} < \varepsilon$$ for all natural numbers $P$ 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.
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"example of summation by parts" is owned by pahio.
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Cross-references: convergent, natural numbers, sum, summation by parts, number, positive, proof, 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 1917 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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