Questions tagged [summation]

11,454 questions
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Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
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How to prove that $\sum\limits_{n=1}^{\infty}n\prod\limits_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a}$?

Mathematica tells me that $$\sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a}$$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
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Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$

I'm looking for a reference with the proof of the following binomial identity: $$\sum_{k=0}^n \binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$$ I've looked in a number of textbooks that have a ...
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Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
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