PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] example of telescoping sum (Example)

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 (“product formula”). E.g. we may write:

$\displaystyle \sum_{k=1}^n\cos{k\alpha} = \frac{1}{\sin\frac{\alpha}{2}}\sum_{k=1}^n\cos{k\alpha}\sin\frac{\alpha}{2}$
The product formula $ \cos{x}\sin{y} = \frac{1}{2}[\sin(x+y)-\sin(x-y)]$ alters this to
$\displaystyle \sum_{k=1}^n\cos{k\alpha} = \frac{1}{2\sin\frac{\alpha}{2}}\sum_{k=1}^n\left(\sin\frac{(2k+1)\alpha}{2}-\sin\frac{(2k-1)\alpha}{2}\right),$
or
$\displaystyle \sum_{k=1}^n\cos{k\alpha} =\frac{1}{2\sin\frac{\alpha}{2}}\left(\... ...alpha}{2}+-\ldots+\sin\frac{(2n+1)\alpha}{2}-\sin\frac{(2n-1)\alpha}{2}\right).$
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.

Bibliography

1
Л. Д. Кудрявцев: Математическийанализ. II том. Издательство ``Высшаяшкола''. Москва(1970).



"example of telescoping sum" is owned by pahio.
(view preamble)

View style:

See Also: goniometric formulas, example of summation by parts


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: de Moivre identity, side, opposite numbers, product formula, terms, sums
There is 1 reference to this entry.

This is version 6 of example of telescoping sum, born on 2007-08-07, modified 2007-08-08.
Object id is 9838, canonical name is ExampleOfTelescopingSum.
Accessed 851 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)