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Old 2021-11-26, 14:13   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

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Default 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 Wikipedia 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.
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