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Does the series $\displaystyle\sum_{n=1}^\infty\dfrac{n\ln(n)+4}{n^2}$ converge or diverge? Which test should be applied? I've tried integral test but I couldn't figure out.

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I have that it diverges. Try splitting the series into the sum of $\frac{nln(n)}{n^2}$ and $\frac{4}{n^2}$ and then test each of those.

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HINT: Note that for all $n$, $nln(n)+4 \gt n$

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$\sum_{n=1}^{\infty} \frac{n\log n}{n^2} =\sum_n\frac{\log n}{n} > \sum_n\frac{1}{n} = \infty$

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