Limit of a sequence, with a sum as the nth element

Question

I need to find the limit of the sequence with a nth element:

$a_n=\sum\limits_{k=1}^{n} \frac{1}{\sqrt{2n^2+k^2}}$

I've seen similar problems that use determined integrals, but I have no idea how to apply them in this case, I would appreciate it if someone could help.

Answer 1

Yes this is Riemann integration, the limit of sum is

$$\lim_{n\to \infty} a_n = \int_{0}^{1} \frac{dx}{\sqrt{2+x^2}}$$

On calculation, we get $\ln(1+\sqrt{3}) - \ln(\sqrt2)$