How to find sum of infinite geometric series with coefficient?Given this series, p + p...

Ashlyn Krause

Ashlyn Krause

Answered

2022-07-23

How to find sum of infinite geometric series with coefficient?
Given this series, p + p ( 1 p ) 3 + p ( 1 p ) 6 + p ( 1 p ) 9 + . . .
This is an infinite geometric series with ratio less than 1 since it's probability.
n = 0 p ( 1 p ) 3 n
Can you use geometric series sum formula? Is it p / ( 1 ( 1 p ) 3 )?
How do you deal with 3 that's in front of n?

Answer & Explanation

Hassan Watkins

Hassan Watkins

Expert

2022-07-24Added 18 answers

Step 1
Your sum is correct. All you need to do is recognize that n = 0 p ( 1 p ) 3 n = n = 0 p [ ( 1 p ) 3 ] n
Step 2
The latter series is a geometric series whose common ratio and leading term are ( 1 p ) 3 and p, respectively, so it will converge to p 1 ( 1 p ) 3 if | ( 1 p ) 3 | < 1 and diverge if | ( 1 p ) 3 | 1

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