<s>[INST] Evaluating $\lim_{n \to +\infty}\frac{1}{n}\left({\frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n}}\right)$ How do I evaluate
$$\lim_{n \to +\infty}\frac{1}{n}\left({\frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n}}\right)\ ?$$
I'm stuck. I tried using the sandwich theorem but I was getting nowhere.
Since $n \lt n+k \lt 2n$, I think it should be $ \infty$ .
 [/INST] using digamma function we have $$\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}=\psi(n+1)-\psi(2n+1)$$
but then using the following asymptotic expansion
$$\psi(x)= \ln(x) - \sum_{k=1}^\infty \frac{B_k}{k x^k}$$
with type two Bernoulli numbers $B_k$, we have 
\begin{align}
\lim_{n \to \infty}\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}&=\lim_{n \to \infty}\Big(\psi(n+1)-\psi(2n+1)\Big)\\
&=\ln2
\end{align}
</s>