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.

  • 2
    $\begingroup$ Hint: The root test. $\endgroup$ – Nobody Jul 22 '16 at 2:11
  • 1
    $\begingroup$ Another hint: Comparison Test... which will lead to the Root Test :) $\endgroup$ – 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.


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