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Watch Hichki (2018) Hd 1080p full movie for free download. Hichki 2018 full movie watch online for free download in hd 1080p.Q:
Why are editors changing the order of summations?
For an easy introduction to the Euler summation formula we usually see the following:
$\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$
The sum $\sum_{k=1}^{n}k$ is one term $\sum_{k=1}^{n}k=k$, which can be multiplied by $k$ and summed up. So we're left with $k(k+1)=k^{2}+k$; $k^{2}$ being the first term and $k$ the second. Now by definition we can sum the first two terms and get a telescopic product to obtain the right hand side.
Here, we see the sum written as $\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$
Is there any particular reason for it? Are the first $k$s really first, and the second $k$s really second? Or is it really the other way around?
A:
The two sums are equivalent, and both may be useful for a given problem.
The two are in fact equivalent: $\sum_{k=1}^n k = \sum_{k=1}^n \underbrace{\left(\sum_{j=1}^k 1\right)}_{\displaystyle \dbinom{k}{j} 0b46394aab
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