Find the sum of the series Σ (2^(n+1)) / (5^n))

[from n=0 to infinity]

Note that after some manipulations, the series turns out be a geometric series.

oo
Σ 2^(n+1) / 5^n.
n=0

oo
Σ (2^n * 2) / 5^n
n=0

..oo
2 Σ (2/5)^n
.n=0

We have a geometric series, where r = 2/5 and a = 2.

2 / (1 - (2/5)) = 2 / (3/5) = 10/3.

The sum of the series is 10/3.

Hope this helped

If you pull a two in front of the sum and change the sum from 1 to inf. (evaluate at n = 0) you get

2 + 2 Σ (2^(n)) / (5^n))

2 + 2 Σ (2/5)^n

2 + 2 (2/5)/(1-2/5)

2 + 2 (2/5)/(3/5)

2 + 2 (2/3)

2 + 4/3 = 3.3333...

2+4/5+8/25+ 16/125+.... , sum=2/(1-2/5) =10/3