# Tagged Questions

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### Relation between $\gcd$ and Euler's totient function .

How to show that $$\gcd(a,b)=\sum_{k\mid a\text{ and }k\mid b}\varphi(k).$$ $\varphi$ is the Euler's totient function. I was trying to prove the number of homomorphisms from a cyclic group of order ...
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### Combinatorial formulas and interpretations

I found that $$\sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
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### Find the sum of the squares of the first $n$ natural numbers

I've been asked to find the sum of the squares of the first $n$ natural numbers. My initial thought was to just program a brute-force solution but I was wondering if there is a mathematical formula ...
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### How can I prove that $(a + b )\oplus(a + c)$ is not possible to simplify. Or is it?

I was trying to simplify the following expression $(a + b )\oplus(a + c)$, where $+$ is just a simple addition of two numbers and $\oplus$ is a binary xor operation. By simplifying I mean exanding or ...
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### Boundary of $\sum_{j}x_j(x_j-x_i)$ for $x_i \in[0,1]$

Does $\sum_{j}x_j(x_j-x_i)$ for $x_i\in[0,1]$ and $0\le i,j\le N-1$ have a upper and lower boundary? And how to calculate them? Thanks!
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### If $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + … + \omega^{n-1} = 0$? [duplicate]

Let $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + ... + \omega^{n-1} = 0$? I saw this on a algebra PPT slice. However the teacher did not explain why this equation is correct, can ...
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### How to calculate $\sum_{n=1}^\infty {p^{n-1}}{(1-p)^{n}} \frac{1}{n} {2n-2 \choose n-1}$?

How to prove $\sum_{n=1}^\infty {p^{n-1}}{(1-p)^{n}} \frac{1}{n} {2n-2 \choose n-1}$ is 1 if $0 \le p \le \frac{1}{2}$ and smaller than 1 if $\frac{1}{2} \lt p \le 1$? I came up with this ...
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### Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Numerical calculations suggest that $$\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,\int_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx=1.553767373413083673460727...$$ How can we prove ...
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### Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
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### How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
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### Evaluate limit of a sum that includes summed term

I am trying to determine whether the limit $$\lim_{n \to \infty}\sum_{k = 2}^{n}\left(\frac{n - k}{n - 2}\right)^{2k} \left(\frac{l - 1}{2}\right)^{k}$$ exists and is finite. No idea how to ...
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### prove simple binomial sum, combinatorics

I want to prove that: $$\large\sum_{i = 1}^{n} \binom{n}{i}\binom{n}{i-1} = \binom{2n}{n-1}$$ On the right hand side we simply have the coefficient of $x^{n-1}$ of the term $(1+x)^{2n}$ But on the ...
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Let $a_i \in \{-1,1\}$ for all $i=1,2,3,...,2014$ and $$M=\sum^{}_{1\leq i<j\leq 2014}a_{i}a_{j}.$$ Find the least possible positive value of $M$. Came across this question in a Math Olympiad and ...
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### Summation of $A\cos (\omega n+\phi)$ [closed]

I'm trying to evaluate the following summation: My original problem is $$\lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N \left|A \cos(\omega n+\phi)\right|^2$$ Now I'm stuck at calculating the ...
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### A strange combinatorial identity: $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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### Jacobian matrix of summation function

So let's say I have a function like this $(\mu_{ij})_{i,j=1,...,t;i+j>t}\longmapsto \sum_{i,j;i+j>t} \mu_{ij}$ and I need to find the Jacobian matrix of that function. I tried to calculate it ...
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### How to prove $\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)}$ diverges?

$$\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)}$$ It seems clear to me that this seires diverges since the dominant term is $(3/2)^n$, a geometric series with $r > 1$ However I am required to ...
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