Does $\sum_{k=1}^{\infty} \frac{1}{k\sqrt{\vphantom{} k+1}}$ converge? Does the following series converge? $$\sum_{k=1}^{\infty} \frac{1}{k\sqrt{\vphantom{|} k+1}}$$ 

I tried using the ratio test and the comparison test but I wasn't able to solve this. 

I think I should try manipulating the denominator to use comparison test but I can't figure out how?
Note that $\frac{1}{k\sqrt{k+1}} &lt; \frac{1}{k \sqrt{k}} = k^{-1.5}$.

Hence, by the comparison test, $\displaystyle\sum_{k=1}^n\frac{1}{k\sqrt{k+1}} &lt; \displaystyle\sum_{k=1}^n k^{-1.5} &lt; \infty$ for all $n$. Hence $\displaystyle\sum_{k=1}^n\frac{1}{k\sqrt{k+1}}$ converges, and in fact it is somewhere  close to $2.04288$ by Wolfram Alpha.
Using the fact that square root of x+1 > square root of (x), for x>0 and p test, it converges