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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.