example of telescoping sum
Some trigonometric sums, as ∑nk=1coskα and ∑nk=1sinkα, 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:
n∑k=1coskα=1sinα2n∑k=1coskαsinα2 |
The product formula cosxsiny=12[sin(x+y)-sin(x-y)] alters this to
n∑k=1coskα=12sinα2n∑k=1(sin(2k+1)α2-sin(2k-1)α2), |
or
n∑k=1coskα=12sinα2(sin3α2-sinα2+sin5α2-sin3α2+-…+sin(2n+1)α2-sin(2n-1)α2). |
After cancelling the opposite numbers we obtain the formula
n∑k=1coskα=sin(2n+1)α2-sinα22sinα2. | (1) |
The corresponding formula
n∑k=1sinkα=-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 identity.
References
- 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 |