# 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.

• Hint: The root test. – Nobody Jul 22 '16 at 2:11
• Another hint: Comparison Test... which will lead to the Root Test :) – KingDuken Jul 22 '16 at 2:14

$$\sum_{n=0}^\infty\frac{1+2^n}{1+5^n} < \sum_{n=0}^\infty\frac{1+2^n}{5^n}=\sum_{n=0}^\infty \frac{1}{5^n} +(\frac{2}{5})^n.$$ Can you take it from here?
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.