mersenneforum.org

mersenneforum.org (https://www.mersenneforum.org/index.php)
-   MattcAnderson (https://www.mersenneforum.org/forumdisplay.php?f=146)
-   -   geometric series (https://www.mersenneforum.org/showthread.php?t=27358)

MattcAnderson 2021-11-26 14:13

geometric series
 
Many of you are familiar with geometric series. Here is a little derivation of a common result.
Finite Geometric Series
Let
S1 = 1 + a + a^2 + ... + a^n.
We multiply S1 by ‘a’ then see
a*S1 = a+ a^2 + … + a^(n+1).
Subtract the second equation from the first one.
(1-a)*S1 = 1-a^(n+1).
Therefore
S1 = [1-a^(n+1)]/(1-a).
We are sure of this. This result about finite geometric series is in many textbooks.
The [URL="https://en.wikipedia.org/wiki/Geometric_series#Sum"]Wikipedia[/URL] on this is very good.
The infinite case is another story.
If S2 = 1 + b + b^2 + … is an infinite sum then
S2 converges for -1<b<1.


All times are UTC. The time now is 02:06.

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.