# example of summation by parts

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

 $\displaystyle\sin{\varphi}+\sin{2\varphi}+\ldots+\sin{n\varphi}=\frac{\sin(n+% \frac{1}{2})\varphi-\sin\frac{\varphi}{2}}{2\sin\frac{\varphi}{2}},$ (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 (http://planetmath.org/ExampleOfTelescopingSum)”. These give the

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

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 (http://planetmath.org/CauchyCriterionForConvergence) 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

 $\left|\sum_{n=N}^{N+P}\frac{\cos{n\varphi}}{n}\right|=\left|\sum_{p=0}^{P}% \frac{1}{N\!+\!p}\cos{(N+p)\varphi}\right|=\left|\sum_{p=0}^{P-1}\left(\frac{1% }{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)S_{N,N+p}+\frac{1}{N\!+\!P}S_{N,N+P}% \right|\leqq$
 $\leqq\sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)|S_% {N,N+P}|+\frac{1}{N+P}|S_{N,N+P}|<$
 $<\sum_{p=0}^{P-1}\left(\frac{1}{N\!+\!p}-\frac{1}{N\!+\!p\!+\!1}\right)\cdot 2% K_{\varphi}+\frac{1}{N\!+\!P}\cdot 2K_{\varphi}\,=\,\frac{1}{N}\cdot 2K_{% \varphi};$

the last form is gotten by telescoping (http://planetmath.org/TelescopingSum) the preceding sum and before that by using the identity

 $S_{N,N+p}=[\cos\varphi+\cos 2\varphi+\ldots+\cos(N\!+\!p)\varphi]-[\cos\varphi% +\cos 2\varphi+\ldots+\cos(N\!-\!1)\varphi].$

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.

Title example of summation by parts ExampleOfSummationByParts 2013-03-22 17:27:56 2013-03-22 17:27:56 pahio (2872) pahio (2872) 8 pahio (2872) Example msc 40A05 ExampleOfTelescopingSum SineIntegralInInfinity ExampleOfSolvingTheHeatEquation