# Thread: Test the series for convergence.

1. ## Test the series for convergence.

Here is it:
$\sum\limits_{n=1}^\infty \frac{ \sqrt{n+1} - \sqrt{n} }{n}$.

2. Originally Posted by TWiX
Here is it:
$\sum\limits_{n=1}^\infty \frac{ \sqrt{n+1} - \sqrt{n} }{n}$.
Use the LCT with the convergent $\frac{1}{n^{3/2}}$

3. Your series will coverge if $\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1$

4. Originally Posted by VonNemo19
Use the LCT with the convergent $\frac{1}{n^{3/2}}$
Failed.
The limit $=\infty$

5. Originally Posted by pickslides
Your series will coverge if $\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1$
I didnt try it.
But its algebric function.
I think it will be 1 ---> Failed.

6. Originally Posted by pickslides
Your series will coverge if $\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1$
That's hard to show, no?

7. $\frac{\sqrt{n+1}-\sqrt{n}}{n}=\frac{1}{n\left( \sqrt{n+1}+\sqrt{n} \right)}<\frac{1}{2n^{\frac{3}{2}}}.$

8. In general not that bad, but this question is a bit sticky yep