Question

How do you find the sum of `2/1+4/3+8/9+16/27+...+2^(n+1)/3^n+...`?

Answer 1

`sum_(n=0)^oo 2^(n+1)/3^n = 6`

Explanation:

`sum_(n=0)^oo 2^(n+1)/3^n = sum_(n=0)^oo (2 xx 2^n)/3^n`

`sum_(n=0)^oo 2^(n+1)/3^n = 2 sum_(n=0)^oo ( 2/3)^n`

This is the sum of a geometric series of ratio: `q=2/3`, and we know that for `q < 1`

`sum_(n=0)^oo q^n = 1/(1-q)`

so:

`sum_(n=0)^oo 2^(n+1)/3^n = 2 (1/(1-2/3)) = 6`