example of summation by parts

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

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

$\sin \phi +\sin 2\phi +\mathrm{\dots }+\sin n\phi ={\displaystyle \frac{\sin (n+\frac{1}{2})\phi -\sin \frac{\phi }{2}}{2\sin \frac{\phi }{2}}},$(1)

$\cos \phi +\cos 2\phi +\mathrm{\dots }+\cos n\phi ={\displaystyle \frac{-\cos (n+\frac{1}{2})\phi +\cos \frac{\phi }{2}}{2\sin \frac{\phi }{2}}},$(2)

proved in the entry “example of telescoping sum (http://planetmath.org/ExampleOfTelescopingSum)”. These give the

$$|\sin \phi +\sin 2\phi +\mathrm{\dots }+\sin n\phi |\leqq \frac{2}{2|\sin \frac{\phi }{2}|}:=K_{\phi },$$

$$|\cos \phi +\cos 2\phi +\mathrm{\dots }+\cos n\phi |\leqq \frac{2}{2|\sin \frac{\phi }{2}|}:=K_{\phi }$$

for any  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$.We want to apply to the series ${\sum }_{n=1}^{\mathrm{\infty }}\frac{\cos n\phi }{n}$ the Cauchy general convergence criterion (http://planetmath.org/CauchyCriterionForConvergence) for series. Let us use here the short notation

$$\cos N\phi +\cos (N+1)\phi +\mathrm{\dots }+\cos (N+p)\phi :=S_{N,N+p}\mathit{\hspace{1em}}(p=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }).$$

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

$$\left|\sum _{n=N}^{N+P}\frac{\cos n\phi }{n}\right|=\left|\sum _{p=0}^P\frac{1}{N+p}\cos (N+p)\phi \right|=\left|\sum _{p=0}^{P-1}\left(\frac{1}{N+p}-\frac{1}{N+p+1}\right)S_{N,N+p}+\frac{1}{N+P}{S}_{N,N+P}\right|\leqq$$

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

$$S_{N,N+p}=[\cos \phi +\cos 2\phi +\mathrm{\dots }+\cos (N+p)\phi ]-[\cos \phi +\cos 2\phi +\mathrm{\dots }+\cos (N-1)\phi ].$$

Thus we see that

for all  natural numbers $P$ as soon as  $N>\frac{2K_{\phi }}{\epsilon }$. According to the Cauchy criterion, the latter series is convergent for the mentioned values of $\phi$. The former series is handled similarly.