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Evaluation of convergence of series $$\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}$$ using camparasion or limit Camparasion Test

What i try

$$\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}\approx\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}}=\frac{5}{7}\sum^{\infty}_{k=1}\frac{1}{k^2}.$$

Which is converge using $p$ series test.

So our original series is converges.

But i have a problem i have just approximate it.can anyone explain me a fair way to do it. Thanks

jacky
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1 Answers1

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It is enough to show that

$$\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}<\frac1{k^2}.$$

  • Thanks Yves daoust. I have a doubt on problem like $\displaystyle \sum^{\infty}_{n=1}\frac{n}{n^2+2}$ which is $\displaystyle \frac{1}{n}$ (Diverges using p series test) but using Divergence test or integral test . It is Diverges.Please explain me. – jacky May 06 '20 at 08:13
  • @jacky: enter another question about this. –  May 06 '20 at 08:17
  • Ok Yves Daoust. – jacky May 06 '20 at 08:18