Calculate the limit of:
$\lim_{n\to\infty}(\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}...\frac{1}{\sqrt{n^2+n^2}})$
Hint: interpret the limit as a sequence of Riemann Sums that converges to a Riemann Integral
I rewrote it as $\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{\sqrt{n^2+i^2}}$ I thought about if I could rewrite it as a Riemann integral but there is no something over n which could define $\delta x$=(b-a)/n. Also, if n tends toward infinity, isn't the sum just 0+0+...+0=0 ? It seems obvious to me that it converges but I don't see how to continue. Thanks for your help !
Hint. One may observe that $$ \frac{1}{\sqrt{n^2+i^2}}=\frac{1}{n}\cdot\frac{1}{\sqrt{1+\frac{i^2}{n^2}}} $$ leading, as $n \to \infty$, to $$ \int_0^1\frac{dx}{\sqrt{1+x^2}} $$ using a Riemann sum.