# Limit comparison test application

I am having some trouble with the following exercise:

I need to determine if the following serie converges or diverges using only the limit comparison test:

$\sum_{n=1}^{\infty} \frac{n}{(4n-3)(4n-1)}$

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I don't know how to proceed.. – Carpediem Nov 29 '12 at 12:19
If you like an answer you could upvote it; you may want to wait a while for some possible future better answers to choose it as "the best answer", but any answer that helps you a little should be, imo, upvoted. – DonAntonio Nov 29 '12 at 12:37

$$\lim_{n\to\infty}\frac{\frac{n}{(4n-3)(4n-1)}}{\frac{1}{n}}=\lim_{n\to\infty}\frac{n^2}{16n^2-16n+3}=\frac{1}{16}\Longrightarrow$$
Do you know the limit comparison test for positive series? It says: let $\,\sum a_n\,\,,\,\,\sum b_n\,$ be positive series s.t. $\,\lim\frac{a_n}{b_n}\,$ exists finitely. Then $\,\sum a_n\,$ converges iff $\,\sum b_n\,$ converges. What's not clear here? – DonAntonio Nov 29 '12 at 12:28