Question

what is the sum of this convergent series: Σ(n=1, ∞) [(sin(3pi/n)+2)/3^n]

Answer 1

What methods are you familiar with? Integral criteria to find bounds? Fourier Transform and Parseval's Identity to find the exact results if boundary functions are known for the Fourier coefficients?

Answer 2

i only know integral >.<, but i can't seem to integrate this hahaha

Answer 3

The only elementary way to solve this integral would be by finding limits, ignore the sin expression, this will give you an integral you can solve and that is strictly smaller than the one you have, this seems like a good approach because you know that:
\[ \large \left| \sin \left( \frac{3 \pi}{n} \right) \right| \leq 1 \]

Answer 4

This will give you a weak lower estimate, which is as good as you can get with elementary analysis for this integral.

Answer 5

whoaa!, thanks a lot >.<

Answer 6

You're welcome