<s>[INST] Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge? 
Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge?

I've tried by using the ratio test but I don't get so far, I'm a little lost with it. Any help will be really aprecciated.
 [/INST] We have $a_n=\frac{2^n+1}{5^n+1}$.
\begin{align*}
\frac{a_{n+1}}{a_{n}}=&\frac{2^{n+1}+1}{2^{n}+1}\cdot\frac{5^{n}+1}{5^{n+1}+1}\\
&=\frac{2+\frac{1}{2^n}}{1+\frac{1}{2^n}}\cdot\frac{1+\frac{1}{5^n}}{5+\frac{1}{5^n}}
\end{align*}
When $n$ tends to $+\infty$, $|\frac{a_{n+1}}{a_{n}}|$ approaches $2/5<1$. Hence the series converges.
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