# Tagged Questions

This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

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### Eigenvalues of the principal submatrix of a Hermitian matrix

This question aims at creating an "abstract duplicate" of various questions that can be reduced to the following: Let $A$ be an $n\times n$ Hermitian matrix and $B$ be an $r\times r$ principal ...
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### Sterling Numbers of The Second Kind With Limitations Placed on Boxes/Parts

I know there are similar problems already on the board. However, none of the previously stated questions contain problems where limitations are placed on the BOXES. Thus, seeing that I am struggling ...
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### How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
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### Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
965 views

### Single Variable Calculus Reference Recommendations

This question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I'm ...
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### Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
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### Different ways of Arranging balls in boxes

This question is generalization of different cases of combinatorics problems that are generally asked. We will find general way of arranging $n$ balls in $r$ boxes. Cases : Identical Balls. ...
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### For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$? [duplicate]

Let $(G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
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### How impact the work of a pure mathematician in the society? [duplicate]

Firstable i explain my situation: On my University most of the careers are doing videos to explain what we do and try to atract more people to our careers. Im in a really bad position, because the ...
5k views

### Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
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### Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$

Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$. I am having trouble showing this. I've seen a similar problem ...
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### a reference for topology [duplicate]

i am looking for a good and easy book about topology that everyone can understand it.also it be interesting.
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### How to determine equation for $\sum_{k=1}^n k^3$

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.
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### Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.