Show that $\sum^\limits{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n}$ diverges

Question

Show that $$\sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n} \ \text{ diverges.}$$

I have tried Root test, Ratio Test, Cauchy condensation Test but all have failed. I think this has to be done by Comparison Test or Limit Comparison Test. But what is the suitable form it has to be reduced to?

Answer 1

Note that $$\dfrac{n-\sqrt{n}}{n^2+5n} \sim \dfrac1n$$ Conclude using limit comparison test.

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EDIT Updated on the request of OP

$$\dfrac{n-\sqrt{n}}{n^2+5n} = \dfrac{1-1/\sqrt{n}}{n+5} = \dfrac1n \underbrace{\left(\dfrac{1-1/\sqrt{n}}{1+5/n}\right)}_{\to 1 \text{ as } n \to \infty}$$

Answer 2

Also you can do in this way:

$$\sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n} > \sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}-n \sqrt n} = \sum^{\infty}_{n=1}\frac{1}{n}$$