can we find $ \lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}} $?

Question

Consider a sequence $ \ a_{n} = \sum_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}} $

Clearly $ \left \{ a_{n} \right \} $ is monotonically increasing and $ a_{n}\leq \frac{n}{\sqrt{n^{2}+n}} \leq 1 \ \ for \ all \ n \epsilon N, \ $ implies $ \lim_{n \to \infty}a_{n} $ exists finitely

We further observe, $$ \frac{n}{\sqrt{n^{2}+n^{2}}} \leq a_{n}\leq \frac{n}{\sqrt{n^{2}+n}} \ \ for \ all \ n \epsilon N $$

$$ => \lim_{n \to \infty}\frac{n}{\sqrt{n^{2}+n^{2}}} \leq \lim_{n \to \infty}a_{n}\leq \lim_{n \to \infty}\frac{n}{\sqrt{n^{2}+n}} $$

$$ => \frac{1}{\sqrt{2}} \leq \lim_{n \to \infty}a_{n} \leq 1 $$

Now, I read in my book $ \lim_{n \to \infty}a_{n} = 2\left ( \sqrt{2}-1 \right )$

How can I attack this problem? Can we really find out this limit?

Answer 1

$\mathcal{Hint}:$ This is equal to

$$ \lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{n\sqrt{1+\frac{k}{n}}}=\int_0^1 \frac{dx}{\sqrt{1+x}}$$

Answer 2

You are trying to find $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{\sqrt{n^2 + kn}}$$

This is equal to $$\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n \frac{1}{\sqrt{1 + \frac{k}{n}}}$$

Consider this as a Riemann sum. Can you finish from here?