\sum_(i=1)^(\infty ) (2)/(3*(5^(i-1)))

Question

finding the infinate sum

Answer 1

∑_{(i = 1 to ∞)} [2/(3(5)^i-1)] = (2/3)[ 1 + 1/5 + 1/25 + 1/125 + ...]

 

The series 1 + 1/5 + 1/25 + 1/125 + ... is geometric with common ratio r = 1/5.  

 

Since -1 < r < 1, the series 1 + 1/5 + 1/25 + ... converges to a₁/(1-r) = 1/(1 - 1/5) = 5/4

 

So,the given series converges to (2/3)(5/4) = 5/6.

Answer 2

a_n = (2/3)5^n-1

The common ratio, 5, is greater than 1.

The series does not have a sum