Find $\sum_{i=1}^n \frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i^2+i}}$

Question

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

I have tried simplifying but I get $$\sum_{i=1}^n \frac{1}{\sqrt{i}}-\sum_{i=1}^n \frac{1}{\sqrt{i+1}}$$ which is subtraction of two divergent series and I don't know where to go from here.

Answer 1

You are almost there. This is a telescoping finite sum. The comment about infinite series is irrelevant because the upper index is $n$, not $\infty$.

We have $$\sum_{i=1}^n \frac{1}{\sqrt{i}} - \sum_{i=1}^n \frac{1}{\sqrt{i+1}} = \sum_{i=1}^n \frac{1}{\sqrt{i}} - \sum_{i=2}^{n+1} \frac{1}{\sqrt{i}} = \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{n+1}},$$ where we have shifted the index of the second summation by $1$.