# Does the series converge or diverge?

## Homework Statement

Determine whether the series $$\sum_{n=1}^{\infty} \frac{1}{n+\sqrt{n}}$$
converges or diverges.

## The Attempt at a Solution

This is driving me nuts. I get nowhere with the root test or the ratio test. I can think of some larger series that diverge (no good) and some smaller series that converge (also no good). What test would you recommend?

The integral test.

n+√n < n+n = 2n

$$\frac{1}{n + \sqrt{n}} > \frac{1}{2n}$$

I'm confused. Doesn't the integral test have us say that if $$\int_{1}^{\infty}{\frac{1}{n+\sqrt{n}} dn}$$ diverges, then our series diverges?

Where do you get the 2n from using the integral test?

Also, the relation [tex]n + \sqrt{n} \leq n+n[\tex] definitely holds, but isn't the series of 1/2n from n=1 to infinity a divergent harmonic series?

Ah, well, I suppose in either case, it's a divergent series. This makes me very happy. Thanks.

I should have said the the comparison test with the integral test . Sorry about any confusion.

1/2n diverges, and since it is greater than 1/(n+√n), the latter also diverges.