Test for series Convergence: $\sum^\infty_{n=1}{\frac {3^n+2^n}{6^n}}$

Question

Does this series converge? $$\sum^\infty_{n=1}{\frac {3^n+2^n}{6^n}}$$ $$\sum^\infty_{n=1}{\frac {3^n+2^n}{6^n}} = \sum^\infty_{n=1}{\frac {3^n}{6^n}+\frac {2^n}{6^n}}=\sum^\infty_{n=1}{\frac {1}{2^n}+\frac {1}{3^n}}$$ Since $\sum^\infty_{n=1}\frac {1}{2^n}$ and $\sum^\infty_{n=1}\frac {1}{3^n}$ both converge, must its sum also converge?

Answer 1

Yes, if $\sum a_n,\sum b_n$ converge to $A$ and $B$ then $\sum (a_n+b_n)$ converges to $A+B$.

Given $\delta>0$ you can find $N_1,N_2$ such that $\sum_{n=1}^Na_n$ is within $\delta/2$ of $A$ if $N>N_1$ and $\sum_{n=1}^Nb_n$ is within $\delta/2$ of $B$ if $N>N_2$.

Now if $N>\max(N_1,N_2)$ you have $\sum_{n=1}^N(a_n+b_n)=\sum_{n=1}^Na_n+\sum_{n=1}^Nb_n$ is within $\delta$ of $A+B$.