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I have the following series: $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n(n + 1)(n + 2)}}$

I must investigate if it is divergent or convergent. I have tried different approaches like the root test, but it didn't help me:(. Could someone point me in the right direction?

Best regards, Petar

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up vote 10 down vote accepted

For large $n$, the $n$th term of your series is roughly $\frac{1}{\sqrt{n}}$. Can you think of another series whose convergence/divergence behavior is known and whose terms can be compared with $\frac{1}{\sqrt{n}}$? Try using the comparison test between your original series and the new series.

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Ah, got it! We know that the harmonic series is divergent. Its members are less than 1/sqrt(n). So if 1/sqrt(n) was convergent also the harmonic series would be convergent. This is a contradiction, so our series is divergent. Thank you again! – Petar Minchev Feb 6 '11 at 8:38
@Petar: You are right. In dealing with convergence questions, it is always useful to try to get a rough idea of what the $n$ term looks like. In cases when you can get a simple approximation for the $n$th term, comparison test is handy. – Dinesh Feb 6 '11 at 8:53
Thanks for the suggestion! This is what I was missing. – Petar Minchev Feb 6 '11 at 8:55

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