<s>[INST] 2 test for convergence problems: $\sum^{\infty}_{k=1}5^k/(3^k+4^k)$and $\sum^{\infty}_{n=1}\tan\left(1/n\right)$ 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
 [/INST] For the first problem 
$$\frac{\frac{5^{k+1}}{3^{k+1}+4^{k+1}}}{\frac{5^k}{3^k+4^k}}$$
$$=\frac{5^{k+1}(3^k+4^k)}{5^k(3^{k+1}+4^{k+1})}$$
$$=5\cdot\frac{\left(\frac34\right)^k+1}{3\cdot\left(\frac34\right)^k+4}$$
So, using Ratio Test:  $$\lim_{k\to\infty}\frac{\frac{5^{k+1}}{3^{k+1}+4^{k+1}}}{\frac{5^k}{3^k+4^k}}=\frac54$$
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