Questions tagged [infinite-product]

For questions on infinite products: convergence, computation, etc...

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How to compute $\displaystyle\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}}$?

In my text book it is stated (without any explanation) that $$\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}} = e^{e^x - 1}$$ and I can't really think of how one can show this.
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If $\alpha$ is the nth root of unity , then the value of [closed]

Given , $\alpha$ is the nth root of unity . Then , the value of $$(11-\alpha)(11-\alpha^2)(11-\alpha^3)........(11-\alpha^{n-1})$$ is equal to ... I tried to use the property : That sum of nth roots ...
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An infinite product

So there is this product $\frac{2}{1} \times \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5}...$ I got its expression of a general term but converting it into a sum by a logarithm doesn't help ...
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Closed form of $f(z) = \prod_{n\in\mathbb Z} \frac{z-n-\bar\alpha}{z-n-\alpha} \times \frac{-z-n-\alpha}{-z-n-\bar\alpha}$

For $\alpha\in\mathbb C$ with $\mathrm{Im}(\alpha)>0$, consider the infinite product $$f(z) = \prod_{n\in\mathbb Z} \frac{z-n-\bar\alpha}{z-n-\alpha} \times \frac{-z-n-\alpha}{-z-n-\bar\alpha}.$$ ...
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Prove that $\prod_{i\in I}X_i\cap\prod_{i\in I}Y_i=\prod_{i\in I}(X_i\cap Y_i)$

So if $X$ and $Y$ are two set then any binary relation between $X$ and $Y$ is a subset of the power set $\mathcal P\big(\mathcal P(X\cup Y)\big)$ of $\mathcal P(X\cup Y)$ so that we can define the set ...
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Evaluate $\lim_{n \to \infty} \prod_{k=n+1}^{2n} k^{1/k}$

Problem: evaluate $\lim_{n \to \infty} \prod_{k=n+1}^{2n} k^{1/k}$. My work: since $n+1 \le k \le 2n$, it is $\frac{1}{2n} \le \frac{1}{k} \le \frac{1}{n+1}$. Since $k \ge 1$, the exponential with ...
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I have this product $$C_t(n)=\prod_{0 < q <t \land q\neq n} \frac{1}{e^{\frac{i\pi q}{t}}-e^{\frac{i\pi n}{t}}}$$ Where $t,n\in\mathbb{N}$. What value will $C_\infty(n)$ tend to? My computer ...