<s>[INST] Show the sequence $a_{n} = \frac{1}{n+1} + \frac{1}{n+2} ... +\frac{1}{2n}$ converges to ln 2 
Possible Duplicate:
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$? 

Please help me with this homework question.
Let $\lbrace a_{n} \rbrace$ be the sequence defined by $$a_{n} = \frac{1}{n+1} + \frac{1}{n+2} ... +\frac{1}{2n}$$
for each positive integer $n$. Prove that this sequence converges to ln 2 by showing that $a_{n}$ is related to the partial sums of the series $\displaystyle\sum\limits_{k=0}^\infty (-1)^{k+1}/k.$
I know that $\displaystyle\sum\limits_{k=0}^\infty (-1)^{k+1}/k$ converges to ln 2.
I also know that
$s_{1} = 1,$
$s_{2} = 1/2,$
$s_{3} = 5/6,$
$s_{4} = 7/12,$
etc.
However, I am having a hard time understanding how $\lbrace a_{n} \rbrace$ is related to the partial sum of the given series. Any help or hints are greatly appreciated.
 [/INST] $$\begin{align*}
1 - \frac{1}{2} &+ \frac{1}{3} - \dots -\frac{1}{2n} = \\
&=1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1} - \left(\frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2n}\right) \\
&=1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1} - \frac{1}{2}\left(1 + \frac{1}{2} + \dots + \frac{1}{n}\right) \\
&=1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1}  + \frac{1}{2}\left(1 + \frac{1}{2} + \dots + \frac{1}{n}\right) - \left(1 + \frac{1}{2} + \dots + \frac{1}{n}\right) \\
&=1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2n}  - \left(1 + \frac{1}{2} + \dots + \frac{1}{n}\right) \\
&=\frac{1}{n+1} + \frac{1}{n+2} + \dots +\frac{1}{2n}\phantom{=}
\end{align*}$$
</s>