Interesting Limit (7)

$\begin{aligned} L&=\lim_{n\to\infty}\sum_{k=1}^{n^2}\frac1 {n^2+k} \\&=\lim_{n\to\infty}\frac1 {n^2}\sum_{k=1}^{n^2}\frac 1 {1+\frac k {n^2}} \\&=\int_0^1\frac {dx}{1+x} \\&=\ln (1+x)\big|_0^1 \\&=\ln 2\approx\boxed{0.69314718} \end{aligned}$

Use the fact that

$\displaystyle \lim_{n\rightarrow \infty}(\sum_{k=1}^n \frac{1}{k} -\ln(n) - \gamma) = 0$

where $\gamma$ is the Euler Constant, the limit we are dealing with is

$\ln (2n^2) - \ln (n^2) = \ln 2$