Download Movies In 720p Khosla Ka Ghosla 1080p !LINK!

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Published: August 1, 2022 (2 weeks ago)
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Download Movies In 720p Khosla Ka Ghosla 1080p !LINK!



 
 
 
 
 
 
 

Download Movies In 720p Khosla Ka Ghosla 1080p

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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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https://colab.research.google.com/drive/1cJweLbwdu4WVYrUV9NPpgP2WMkEUdmbi

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