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Prove that if $n > 3$ is odd, then $\frac{1}{n} +\frac{1}{n+1} +\frac{1}{n+2}+ \cdots + \frac{1}{n+n-1} = \frac{2}{n}$.

Prove that if $n > 3$ is odd, then $\dfrac{1}{n} +\dfrac{1}{n+1} +\dfrac{1}{n+2}+ \cdots + \dfrac{1}{n+n-1} = \dfrac{2}{n}$.
I tried to multiply the LHS by $n-1$ and then use the argument principle to find some poles. But I can’t find a pole. Can you please give me a hint? Thanks in advance.

A:

This is a trick used by Euler to prove the formula
$$\sum_{k=0}^{n-1} \frac{1}{k+1} = \frac{n}{n+1}.$$
In order to prove it, let $f(z) = \frac{1}{z} + \frac{1}{z+1} + \frac{1}{z+2} + \cdots + \frac{1}{z+n-1}$ be a rational function, and set $f(n) = \sum_{k=0}^{n-1} \frac{1}{k+1}$ and $z = n$. Then we have
$$ f(n) = \frac{1}{n} + \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n-1},$$
so that
\begin{align*}
f(n) – \frac{1}{n} &= \sum_{k=0}^{n-2} \frac{1}{k+1} + \frac{1}{n} – \frac{1}{n} \\
&= \sum_{k=0}^{n-1} \frac

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