I want to find the sum of the following series:
$$\sum_{n=1}^{\infty}{\frac{2^{n+1}-8}{3^n}}$$
I have the intuition that this is a geometric series judging by the $n$ exponent in the numerator and denominator. Therefore I try to simplify a little bit:
$$\sum_{n=1}^{\infty}{\frac{2\cdot2^n-2^3}{3^n}}=\dots$$
I'm stuck here… I know that I need to group something but I'm not sure how. Preferably I want to end up with something like $\left(\frac{2}{3}\right)^n$.
Any hints on how to proceed to find the sum of the geometric series (which I know is equal to $\frac{1}{1-q}$ for $n=0$?
Best Answer
Hint:
First, use the fact that $$\frac{a+b}{c} = \frac ac + \frac bc$$
and then, use the fact that if $$\sum_{i=1}^\infty a_n\\\sum_{i=1}^\infty b_n$$ both converge, then $$\sum_{i=1}^\infty (a_n+b_n) = \sum_{i=1}^\infty a_n + \sum_{i=1}^\infty b_n$$
Additional facts to use: