Evaluate $\lim\big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\big)$ using sequential methods 
Evaluate $$\lim\Big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\Big)$$ using sequential methods.

Of course: $$\frac{1}{n}\bigg(\frac{1}{1+1/n}+\frac{1}{1+2/n}+\ldots+\frac{1}{1+n/n}\bigg) \rightarrow \int_{0}^{1}\frac{dx}{1+x}=\ln 2$$
but by using only sequences, I don't know where to start from. I thought of the squeeze theorem, but i am not sure how to get $\ln 2$ on the way. 
Thanks in advance.

We have 
$$\sum_{k=1}^n \frac1{n+k}=\frac1n\sum_{k=1}^n \frac1{1+k/n}=\frac1n\sum_{k=1}^n\sum_{j=0}^\infty (-1)^j\left(\frac k n\right)^j=\ldots$$
and for any fixed $n$ since by Faulhaber's formula 
$$\sum_{k=1}^n k^{j} = \frac{n^{j+1}}{j+1}+O(n^j)$$
we have
$$\ldots =\sum_{j=0}^\infty \frac{(-1)^j}{n^{j+1}}\sum_{k=1}^nk^j
= \sum_{j=0}^\infty \frac{(-1)^j}{j+1}+O\left(\frac1n\right)\sum_{j=0}^\infty (-1)^j$$
$$\to\sum_{j=0}^\infty \frac{(-1)^j}{j+1}=1-\frac12+\frac13-\frac14+\ldots=\ln 2$$
indeed by Alternating harmonic series the result follows.
