# Tagged Questions

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### Evaluating a limit involving summation [duplicate]

Evaluate : $$\lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right)$$ Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried ...
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### Sum of Logarithms of expoenetial functions

Given a matrix $P \in \Re^{n \times d}$, and a column vector $\theta \in \Re^d$. Assume that $\sum\limits_{i=1}^n \ln{(1+e^{P_i\theta})} \leq 1$, where $P_i$ is the $i^{th}$ row in $P$. What can be ...
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### Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$\ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1$$ I ask this question as I'm ...
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### Is there a way to iterate through a set?

I have a set $X=\{x_1,x_2,...x_n\}$ and I want to define a function: $$f(X)=\prod_{j=1}^n{\sum_{i=j}^nx_i \choose x_j}$$ However, in this function I'm treating this set as a sequence, as sets don't ...
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### Even harmonic sums? [duplicate]

How do we calculate this? $\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{2n}}{2n}$ I am stuck that the integrals isn't converging for harmonic (even) numbers . somebody please help .
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Sum of Independent Levy RVs is Levy RV [on hold]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
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### Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry ...
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### Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
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### Evaluating scalar functions of vectors in multidimensional simplices part II

In this question we want to generalize the result from Evaluating scalar functions of vectors in multidimensional simplices. . To be precise we consider a following multivariate sum: {...
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### $\sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} =?$ [on hold]

Can anyone sum up this series? $f(z, t) = \sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} .$
### 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 ...