Prove that $\sum\limits^\infty_{n=0}\frac{3n^2+2} {n \cdot (5n-1)^2}$ is convergent How can I prove that $\sum\limits^\infty_{n=0}\frac{3n^2+2} {n \cdot (5n-1)^2}$ is convergent? I failed doing it with the comparison test and root test aswell as fraction test seem far too complicated to me. Is there an easy way?
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$$\frac{3n^2+2}{n(5n-1)^2}\ge\frac{3n^2}{n\cdot25n^2}=\frac3{25}\cdot\frac1{n}$$
See Limit Comparison Test. As $a_n$ is of order $1/n$ the series will diverge.