Stuck on Algebra for Summation Problem

Question

I want to find what the exact value of this sum is. What I was given is: $$\sum_{i=1}^n \frac{1}{\sqrt{2i-1} + \sqrt{2i+1}}$$

The only thing I can think of is turning the denominator into the form:

$$\sum_{i=1}^n \frac{1}{(2i-1)^{\frac{1}{2}} +(2i+1)^{\frac{1}{2}}}$$

I was wondering if I could get a hint on what to do next.

Answer 1

\begin{align*} \sum_{i=1}^n \frac{1}{\sqrt{2i-1} +\sqrt{2i+1}}&=\frac12\sum_{i=1}^n\left(\sqrt{2i+1}-\sqrt{2i-1} \right)\\ &=\frac12(\sqrt{2n+1}-1).\\ \end{align*}