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I am having a problem with determining wether this series is convergent or divergent:

$$\sum\limits_{n=0}^\infty \frac{n}{(4n-3)(4n-1)}$$

I just started the chapter on series and we've learned various convergence tests.

Thank you in advance

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1 Answer 1

The terms don't go to $0$ fast enough for convergence.

Note that for $n\ge 1$, the $n$-th term is $\gt \dfrac{n}{(4n)(4n)}$, that is, $\gt \dfrac{1}{16}\cdot \dfrac{1}{n}$.

But the harmonic series diverges.

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