# Questions tagged [infinite-product]

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

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### Question on convergence of infinite product [duplicate]

Show that $\prod_{n=1}^{\infty}(1+\frac{(-1)^n}{\sqrt{n}})$ is divergent. Clearly the first term is 0, but the rest are positive real numbers. My textbook defines convergence even when a series ...
1 vote
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### Definite integral over an infinite product

Evaluate the following integral $$\int_0^\infty\frac{x+1}{x+2}\cdot\frac{x+3}{x+4}\cdot\frac{x+5}{x+6}\cdots dx$$ When I saw this, I was pretty sure that the infinite term must telescope or it must ...
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### References for the Euler function $\prod_{k=1}^\infty{(1-q^k)}$

I was thinking about probability games when I stumbled upon an interesting sequence, namely $$\prod_{k=1}^\infty{(1-q^k)} = (1-q)(1-q^2)(1-q^3)...$$ I was specifically interested in a proof on whether ...
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### Calcualte the limit of $\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)*...*(1+\sqrt n)}$ as n approaches infinity [duplicate]

$\lim_{n \to +\infty}\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)*...*(1+\sqrt n)}$ Any hints / solutions here? I feel like I'm stuck. The Ratio test is equal to $1$ which doesn't help at all. Intuitively, ...
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### An infinite sum of products

I have to calculate this sum in closed form $$\sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k}$$ where $x < 1$. Numerical evaluation shows that this converges. The product can be performed ...
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### What is $x^\bot$? Is $\zeta(\bot)=\bot$ for Riemann's zeta function $\zeta$ and wheel theory's $\bot$?

Background: A wheel is an algebraic structure $(W,0,1,+,\cdot, /)$ where: $W$ is a set, $0,1\in W,$ $+$ and $\cdot$ are binary operations, $/$ is a unary operation, and $+,\cdot$ are associative, ...
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