Find the value of the summation from n equals 2 to infinity of the quotient of the sum of the quantity 3 to the nth power and 5 to the nth power and the quantity 15 to the nth power

Question

Find the value of  

Type your answer in the space below and give 2 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.48).

Answer 1

(3ⁿ + 5ⁿ)/15ⁿ = (1/5)ⁿ + (1/3)ⁿ.

So, we have sum of 2 infinite geometric series: S₁ = ∑₂^∞(1/5)ⁿ = (1/25)/(1 - 1/5) = (1/25)/(4/5) = 1/20;

S₂ = ∑₂^∞(1/3)ⁿ = (1/9)/(1 - 1/3) = (1/9)/(2/3) = 1/6

Then, ∑₂^∞ (3ⁿ + 5ⁿ)/15ⁿ = 1/20 + 1/6 = 13/60