# Tagged Questions

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### Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
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### Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (...
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### Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
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### In how many ways can the integers from $1$ to $n$ be divided into two groups with the same sum?

In how many ways can the integers $1,2,\ldots,n$ be divided into two groups with the same sum? I have tried calculating some of these values for small $n$, but cannot seem to find a pattern. Any ...
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### How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
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### Why use Einstein Summation Notation?

Einstein summation convention dictates that repeated indices should be summed. Thus the equation $a_{ij} = b_{ik}c_{kj}$ is taken to mean $a_{ij} = \sum_k b_{ik}c_{kj}$ where in both cases the range ...
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### A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
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### A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$H_n=\sum_{k=1}^{n}\frac{1}{k},$$ and the generalized harmonic numbers are defined by $$H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}.$$ Recently, I have found ...
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### How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$\sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
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### Subset Sum Problem Variation?

There are $100$ cards with a unique number from $1$ to $100$ written over them. How many ways can someone pick exactly $5$ cards where the numbers on them sum to $100$? I am not sure but this could ...
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### Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx.$$ Here is what I tried (but I do not think this is ...
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### Series with $k^2$ coefficients

Let $\{S_k\}$ be a sequence of numbers. Then by reversing the sum we have the following, $$A=\sum_{k=0}^n k\cdot S_k \cdot S_{n-k} = \sum_{k=0}^n (n-k) \cdot S_{n-k} \cdot S_{k} .$$ Thus adding these ...