Find the sum of the infinite series (-1 + (-3)^n) / 4^n from n =2 to infinity

Question

Find the sum of the infinite series (-1 + (-3)ⁿ) / 4ⁿ starting from n = 2 all the way to infinity. I figured it would be a geometric series, but I am having trouble finding the ratio in order to determine the sum.

Answer 1

I would think of it as a sum of fractions with the same denominator. ∑ -(1/4)^n + (-3/4)^n and then break it into a sum of sums ... ∑ -(1/4)^n + ∑ (-3/4)^n which do converge and will easily add together.

Answer 2

Try separating into two separate infinite series:

-1 ( -3)ⁿ

--- + --------

4ⁿ 4ⁿ

Starting with n = 2, the first series becomes: -1/16 + -1/64 + ... (r = 1/4, a = -1/16)

, the 2nd series: 9/16 + (-27/64) + .... (r = -3/4, a = 9/16)

Since |r| < 1 in both cases use S = a/(1-r) to get the sum of each.