Questions tagged [infinite-product]

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

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Interesting Infinite Product Involving Subfactorials (Dearrangements)

I'm trying to compute: \begin{gather*} P =\lim_{n \to \infty} \prod_{i = 2}^n \left ( 1 - \frac{(-1)^{i-1}}{S_i} \right ) \end{gather*} Where $S_i$ is given by: \begin{gather*} i!\sum_{j = 1}^{i-1} \...
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The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?

I am looking for a closed form for $R=\frac12\exp\int_0^1-\log\left(\sin\left(\frac{\pi}{6}+\frac{2\pi}{3}x\right)\right)\mathrm dx\approx0.6159$. Wolfram does not give a closed form for $R$. Wolfram ...
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Why $\prod\limits_{n=1}^{\infty} \biggl (\phi(q^n)^{\mu(n)} \biggr)= 1-q$?

Playing with Euler $\phi$ function (not to be confused with the totient function, here another reference), I found this curious identity (I calculated it for various $q$ with Mathematica and it holds)...
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Expressing $\prod_{n=1}^{\infty}\left(1-\frac{x}{n^{2}}\right)$ as an elementary function [closed]

How do you express the infinite product: $$\prod_{n=1}^{\infty}\left(1-\frac{x}{n^{2}}\right)$$ as an elementary function?
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Analysis of a deceptively simple infinite product: $f(n) = \prod_{j=2}^{\infty}\left(1-\frac{1}{j^n}\right)$

I am reading the book "An introduction to Infinite Product" by Charles H. C. Little , Kee L. Teo and Bruce van Brunt (https://doi.org/10.1007/978-3-030-90646-7). The book mentions the ...
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Powers of 5 in an infinite product series [closed]

Consider the highest power of 5 in the product $1^1. 2^2 . 3^3 .....n^n$ is given by $H(n)$ (i.e., $H(n)$ denotes the largest integer $k$ such that $5^k$ is an integral dvisior of the above product. ...
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Limit of a product of infinite terms given by splitting into infinite limits

I came across the following limit of product of infinite terms. But the solutions given in the web was by taking separate limit of each term of the infinite terms and taking the product. I have learnt ...
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A coin lands on heads with probability $\frac1{9n+1}$ on $n-$th toss. What’s the probability it will land on heads eventually?

Question Suppose there is a coin which lands on heads with probability $\frac1{9n+1}$ on $n-$th toss. What’s the probability it will land on heads eventually? Reasoning The answer can be given by the ...
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Functional Equation for Holomorphic Infinite Product

Letting the lattice $\Lambda := \{w_{mn} := m + in : m,n \in \mathbb{Z} \}$, let $H(z) := z \prod_{w \in \Lambda } (1 - \frac{z^4}{w^4})$. I am trying to show that $H(z-1) = -e^{-\pi(z-1)/2}H(z)$, ...
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How can we formally / rigorously use Mertens' third theorem with $n^2 - 1$ instead of $\ln n$?

I'm quite new to Analytic NT, so was wondering if the following is true, and beyond being obviously true, how could we prove it rigorously line-by-line? See: Mertens Third theorem The function they ...
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