Limits of sequence Riemann sums

Question

I'm trying to calculate the limit by interpreting the limit as a sequence of Riemann sums that converges to the Riemann integral:

$$ \lim\limits_{n \to + \infty} \sum\limits_{k=1}^n \frac{1}{\sqrt{n^{2}+2kn}} $$

I have:

$$ \lim\limits_{n \to + \infty} \sum\limits_{k=1}^n \frac{1}{\sqrt{n^{2}+2kn}} = \lim\limits_{n \to + \infty} \frac{1}{n} \sum\limits_{k=1}^n \frac{1}{\sqrt{1+\frac{2kn}{n^{2}}}} $$

but don't know how to get to the integral and compute the limit. Any ideas?

Answer 1

You should write:

$$ \lim\limits_{n \to + \infty} \frac{1}{n} \sum\limits_{k=1}^n \frac{1}{\sqrt{1+\frac{2kn}{n^{2}}}}= \lim\limits_{n \to + \infty} \frac{1}{n} \sum\limits_{k=1}^n \frac{1}{\sqrt{1+2\frac{k}{n}}}$$

Do you see the integral now? It is:

$$\int_0^1 \frac{1}{\sqrt{1+2x}}\,dx$$