Prove that X n=0 to ∞ sin(2n)/1 + 5^n is absolutely convergent. (PLS explain clearly and more)

Question

Prove that X n=0 to ∞ sin(2n)/1 + 5^n is absolutely convergent.

Answer 1

Not sure whether you are looking for the sequence of the series.

The sequence clearly goes to zero since |sin 2n|≤1 and the denominator is unbounded.

The series (sum) converges (absolutely) by the comparison test since since

|sin 2n|/(1+ 5ⁿ| ≤1/(1+5ⁿ) which clearly converges.