# Investigate convergence of the following series: <munderover> &#x2211;<!-- ∑ --> <mrow c

Investigate convergence of the following series:
$\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{2+\left(-1{\right)}^{n}}{\pi }\right)}^{n}$
Which convergence criterion shoul be applied?
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lorienoldf7
The series is absolutely convergent by comparison with the geometric series with ratio $\frac{3}{\pi }$
In particular:
$\sum _{n\ge 0}{\left(\frac{2+\left(-1{\right)}^{n}}{\pi }\right)}^{n}=\sum _{n\ge 0}{\left(\frac{9}{{\pi }^{2}}\right)}^{n}+\sum _{n\ge 0}\frac{1}{{\pi }^{2n+1}}=\frac{{\pi }^{2}}{{\pi }^{2}-9}+\frac{\pi }{{\pi }^{2}-1}.$