TITLE: 2 test for convergence problems: $\sum^{\infty}_{k=1}5^k/(3^k+4^k)$and $\sum^{\infty}_{n=1}\tan\left(1/n\right)$
QUESTION [6 upvotes]: For the first problem: $$\sum^{\infty}_{k=1}\frac{5^k}{3^k+4^k}$$
I tried to take the ratio test but was having trouble simplifying $$\frac{3^k + 4^k}{3^{k+1} + 4^{k+1}}$$
For the second problem: $$\sum^{\infty}_{n=1}\tan\left(\frac{1}{n}\right)$$
I don't know how to approach it. 
Thanks for the help as always

REPLY [3 votes]: Note that $$\lim_{n\to\infty}n^{+1}\tan(n^{-1})=1<\infty$$ so the corresponding series is divergent. That is similar to the following series: $$\sum_1^\infty\sin\left(\frac{1}n\right)$$