# example of telescoping sum

Some trigonometric sums, as $\sum_{k=1}^{n}\cos{k\alpha}$ and $\sum_{k=1}^{n}\sin{k\alpha}$, may be telescoped if the terms are first edited by a suitable goniometric formula (http://planetmath.org/GoniometricFormulae) (‘‘product formula’’). E.g. we may write:

 $\sum_{k=1}^{n}\cos{k\alpha}\;=\;\frac{1}{\sin\frac{\alpha}{2}}\sum_{k=1}^{n}% \cos{k\alpha}\sin\frac{\alpha}{2}$

The product formula  $\cos{x}\sin{y}\,=\,\frac{1}{2}[\sin(x\!+\!y)-\sin(x\!-\!y)]$  alters this to

 $\sum_{k=1}^{n}\cos{k\alpha}\;=\;\frac{1}{2\sin\frac{\alpha}{2}}\sum_{k=1}^{n}% \left(\sin\frac{(2k\!+\!1)\alpha}{2}-\sin\frac{(2k\!-\!1)\alpha}{2}\right),$

or

 $\sum_{k=1}^{n}\cos{k\alpha}\;=\;\frac{1}{2\sin\frac{\alpha}{2}}\left(\sin\frac% {3\alpha}{2}-\sin\frac{\alpha}{2}+\sin\frac{5\alpha}{2}-\sin\frac{3\alpha}{2}+% -\ldots+\sin\frac{(2n\!+\!1)\alpha}{2}-\sin\frac{(2n\!-\!1)\alpha}{2}\right).$

After cancelling the opposite numbers we obtain the formula

 $\displaystyle\sum_{k=1}^{n}\cos{k\alpha}\;=\;\frac{\sin\frac{(2n+1)\alpha}{2}-% \sin\frac{\alpha}{2}}{2\sin\frac{\alpha}{2}}.$ (1)

The corresponding formula

 $\displaystyle\sum_{k=1}^{n}\sin{k\alpha}\;=\;\frac{-\cos\frac{(2n+1)\alpha}{2}% +\cos\frac{\alpha}{2}}{2\sin\frac{\alpha}{2}}.$ (2)

is derived analogously.

Note.  The formulae (1) and (2) are gotten also by adding the left side of the former and $i$ times the left side of the latter and then applying de Moivre identity.

## References

• 1 Л. Д. Кудрявцев: Математический анализ. II том.  Издательство  ‘‘Высшая школа’’. Москва (1970).
Title example of telescoping sum ExampleOfTelescopingSum 2013-03-22 17:27:21 2013-03-22 17:27:21 pahio (2872) pahio (2872) 11 pahio (2872) Example msc 40A05 GoniometricFormulae ExampleOfSummationByParts DirchletKernel