<s>[INST] Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$ Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$
So if i apply ratio test, I get $\lim_{x\to \infty} 5|x||\frac{n+2^n}{n+1+2^{n+1}}| $
Now need help to to check endpoints x= 2/5 and -2/5 
If i put in x= 2/5 i get $\sum_{n=1}^\infty$ $\frac{2^n}{n+2^n}$ How do i check if this converges or not? Same with x = -2/5 $\sum_{n=1}^\infty$ $\frac{(-2)^n}{n+2^n}$
 [/INST] By ratio test
$$ 5|x||\frac{n+2^n}{n+1+2^{n+1}}|\sim_\infty 5|x||\frac{2^n}{2^{n+1}}|=\frac{5}{2}|x|<1\iff|x|<\frac{2}{5}$$
hence the radius is $\frac{2}{5}$.
Added
-For $x=\frac{2}{5}$, $$\frac{5^n}{n+2^n}x^n=\frac{2^n}{n+2^n}\to1\neq0$$
so the series is divergent.
-For $x=-\frac{2}{5}$,
$$\frac{5^n}{n+2^n}x^n=\frac{(-1)^n2^n}{n+2^n}$$
has not a limit so the series is also divergent.
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