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Published: August 1, 2022 (2 weeks ago)
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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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