example of telescoping sum

Some trigonometric sums, as k=1ncoskα and k=1nsinkα, may be telescoped if the terms are first edited by a suitable goniometric formulaPlanetmathPlanetmath (http://planetmath.org/GoniometricFormulae) (‘‘product formula’’). E.g. we may write:


The product formula  cosxsiny=12[sin(x+y)-sin(x-y)]  alters this to




After cancelling the opposite numbers we obtain the formula

k=1ncoskα=sin(2n+1)α2-sinα22sinα2. (1)

The corresponding formula

k=1nsinkα=-cos(2n+1)α2+cosα22sinα2. (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 identityMathworldPlanetmath.


  • 1 Л. Д. Кудрявцев: Математический анализ. II том.  Издательство  ‘‘Высшая школа’’. Москва (1970).
Title example of telescoping sum
Canonical name ExampleOfTelescopingSum
Date of creation 2013-03-22 17:27:21
Last modified on 2013-03-22 17:27:21
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Example
Classification msc 40A05
Related topic GoniometricFormulae
Related topic ExampleOfSummationByParts
Related topic DirchletKernel