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I have this series and I am trying to determine if it is convergent or divergent. I already tried the comparison test and it seems to fail but only perhaps because I am breaking apart the sequence incorrectly.

$$\sum_2^\infty \frac{n+30}{n^2\sqrt n}.$$

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  • $\begingroup$ limit comparison test? $\endgroup$ – user99914 Nov 23 '15 at 5:17
  • $\begingroup$ use the universal convergence test $\endgroup$ – Thoth Nov 23 '15 at 5:19
  • $\begingroup$ You can divide top and bottom by $n$, then use direct comparison. $\endgroup$ – James Nov 23 '15 at 5:21
  • $\begingroup$ I triedthe limit comparison test but perhaps by setting b(subn) as $\frac{n}{n^{2}\sqrt{n}}$ is not correct? $\endgroup$ – Erik Nov 23 '15 at 5:21
  • $\begingroup$ For limit comparison your $b_n$ is fine. Maybe more simply observe that the top is $\le 31n$. $\endgroup$ – André Nicolas Nov 23 '15 at 5:27
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It seems to me that you should forget about the thirty in the numerator, divide top and bottom by the n in the numerator, then by the Direct Comparison Test with the p-series with p=3/2 it follows that the series converges.

Check out Paul's Online Notes: Series Strategies

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