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I need to determine whether this series converge or not - $$\sum^\infty_{n=1} \frac{^4\sqrt n-\ 4}{n^\frac{3}{2}+6}$$ I would like to get an hint - I thought about the ratio test , but it seems too long.

Thanks!

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    $\begingroup$ Looks an awful lot like $\root 4\of n/ n^{3/2}$ to me ... $\endgroup$ – David Mitra Apr 15 '16 at 11:56
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Have you tried the comparison test? This function seems ugly to do at any rate...

If you compare with the sum of $\frac{n^{\frac{1}{4}}}{n^{\frac{6}{4}}}=\frac{1}{n^{\frac{5}{4}}}$, which is a p series with $p>1$ and so converges.

EDIT: Rather, you should note that each term of your infinite series is less than each term of the $\frac{1}{n^{\frac{5}{4}}}$ series, which converges, and so the original series converges.

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  • $\begingroup$ Can I use the comparison test while the series isn't positive for all n ? $\endgroup$ – Barak Mi Apr 15 '16 at 12:12
  • $\begingroup$ @BarakMi You can always can if the series sequence is positive for almost all $\;n\;$ $\endgroup$ – DonAntonio Apr 15 '16 at 13:30
  • $\begingroup$ and my explanation should be that for some n the series is always positive? $\endgroup$ – Barak Mi Apr 15 '16 at 13:31
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    $\begingroup$ Think about it this way. If $n>4^4$, then DCT holds for those terms. Consequently the infinite series beginning at $n=4^4$ and going to infinity converges to a finite value. Now note that the sum from 1 to $n=4^4$ is already finite. The sum of two finite quantities is finite therefore the infinite series is finite; that is, it converged $\endgroup$ – KR136 Apr 15 '16 at 15:44

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