Tagged Questions

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Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
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Integral of a summation related to $\sin$ expansion

I am trying to evaluate the following integral. It has similarity to the Maclaurin expansion for $\sin$. $$\int_{-\infty}^\infty{\sum_{n=0}^{\infty}\frac{(-1)^n}{\left(2n+x^2\right)!}}\text{dx}$$ ...
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For $\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$, show there is a choice of signs such that it is irrational [on hold]

Considering $$\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$$ where you can replace each $\pm$ with $+$ or $-$. Prove that there is at least one choice of signs such that the number is ...
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$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [on hold]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
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Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
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Bound on binomial summation

The bound for $\sum_{i=1}^n\binom{n}{i}2^i$ is $O(3^n)$ but what will be the bound for $\sum_{i=1}^{\frac{n}{2}}\binom{n}{i}2^i$. Any idea how should I proceed.
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Order of summation for shifted exponential function

I want to represent the function: $$f(x)=e^{-a(x-b)^{2}}$$ where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$. As a power series for an integral I am working ...
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GCD Summation function

I know that GCD summation function ($\sum_{i=1}^{i=n} \gcd(i, n)$ is multiplicative. Thus it can be calculated in $O(\log n)$ complexity using factorization. But I want to want to compute the same ...
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Bizarre binomial sum

It is many times that we need to compute discrete convolutions. Driven by this need we have discovered a following formula: \sum\limits_{l=0}^k \binom{l+A_1}{A_2} \binom{-2 \beta (k-l)...
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How to prove that a sum of $\cosh(kx)$ is equal to a formula? [duplicate]

I need to prove that $$\sum_{k=0}^{n}\cosh(kx) = \frac{\sinh((n+1/2)x) + \sinh(x/2)}{2\sinh(x/2)}$$ Can you help me out? How do I even start?
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Closed form for $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n^2+a^2}}$

Do the convergent sum $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n^2+a^2}}$$ posses a closed form? ($a \in \mathbb{R}$) Special case is known, for $a=0$ one recalls well known alternating harmonic ...
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Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
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Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 - \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by ...
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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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How to find$\sum_{i,j,k\in \mathbb{Z}}\binom{n}{i+j}\binom{n}{j+k}\binom{n}{i+k}$ for $n \in \mathbb{N}$

Yeah, it's $$\sum_{i,j,k\in \mathbb{Z}}\binom{n}{i+j}\binom{n}{j+k}\binom{n}{i+k}$$ and we are summing over all possible triplets of integers. It appears quite obvious that result is not an infinity. ...
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double summation of conditional variable depending on sum of integer

I am having trouble with taking a certain summation and finding an explicit value for the summation. The summation is: $$S = \sum_{w=3}^a \lambda_w \sum_{m=w}^a \lambda_m$$ The only information ...
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Analytic Continuation of Sum $\sum_{n=0}^{\infty} e^{-b \sqrt n}$

Suppose we have the following function: $$f(b)=\sum_{n=0}^{\infty} e^{-b \sqrt n}$$ Is there a closed form expression for the analytic continuation of $f(b)$ to $f(-b)$?
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Multivariate sum over a simplex

Let $s\ge 0$ and $d \ge 0$ be integers. Let $\left\{ a_\eta \right\}_{\eta=0}^s$ be some positive numbers and let $0 \le \xi_1 \le \xi_2 \le \cdots \le \xi_d \le s$. Finally let $k$ be a strictly ...
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Associative notation for Sigma notation

I wonder the following equation does make sense or possible: $(\sum_{k=0}^{L}-\sum_{k=L+1}^{2L})y[k]$ Instead of: $\sum_{k=0}^{L}{y[k]}-\sum_{k=L+1}^{2L}{y[k]}$ In my opinion, advantage of this is ...
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