# Tagged Questions

181 views

### Intuitive ways to get formula of binomial-like sum

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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### does anyone know a nice form of the infinite sum $\sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}$?

I was wondering if anyone on here knows of a closed form or special function for this infinite sum: $$\sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}$$ Or the sum of any non-trivial subset.
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### Series expansion of {x}

Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ...
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### Check if the sum is equal to the polynomial

I have the following polynomial $$(1-\alpha)+3\alpha\beta\gamma+4\alpha\beta\gamma[(1-\beta)+(1-\gamma)]+5\alpha\beta\gamma[(1-\beta)^2+(1-\beta)(1-\gamma)+(1-\gamma)^2]+\cdots$$ I believe I can ...
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### Show that $\sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2$

Following from the question here, I was wondering if it's possible show directly that $$\sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2$$ without expanding the summation in full on either side.
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### How to get analytical summation of this series?

How to get the analytical summation of this series? $$\sum\limits_{n = 2}^{ + \infty } {{\varepsilon ^{n - 1}}\frac{1}{{{n^3}}}\frac{{{d^2}P_n^2\left( {\cos \theta } \right)}}{{d{\theta ^2}}}} = ?$$ ...