# Questions tagged [infinite-product]

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

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### Infinite product expansion of $\frac{1}{\ln(x)}$?

It's well-known that \begin{array}{l}\frac1{1-x}=\sum_{n=0}^\infty x^n=\prod_{k=0}^\infty{\textstyle\sum_{i=0}^{m-1}}x^{im^k}.\\\end{array} By using the following "theorem about product and sum&...
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### Solutions of $((a-1)!)^x=0$

I have an equation that I couldn't solve. I will be glad if you help me to do it. Are there solutions in: $((a-1)!)^x=0,$ with $a\in\mathbb N?$ Because with my knowledge in math, I found no solutions.
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### e to an integral as an infinite product via the definition of the integral

Is there a name for this relationship? I am having a hard time searching for it. I'm hoping I've typed this up correctly, it's my first question here. $x_k^\star$ is for instance $\frac{k \cdot h}{n}$ ...
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### Convergence of series $\sum \arcsin\left(\frac{r_n \sin(\theta_n)}{\sqrt{1+2r_n\cos(\theta_n)+r_n^2}}\right)$

Let $r_n$ be a sequence of strictly positive numbers converging to $0^+$. Let $\theta_n$ be a sequence of values in $[0,\tfrac{\pi}{2}]$ which may or may not converge. Consider the following sequence ...
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### Approximate result for $\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)$?

What would be a quick way to approximately determine the value of $$\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)=\phi\left(\frac{1}{2}\right) ,$$ where $\phi(q)$ is the Euler function? By ...
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### Convergent Infinite Product in Proof of Partitions Identity

I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998). Theorem 1.1 is a standard result that writes the generating function of a ...
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### How do I prove this equation? $\prod _{n=0}^j(1+x^{^{2^n}})=\sum _{m=0}^{2^j+1}x^m$

So I just stumbled upon this while trying to find the limit of a series. I tried induction but didn't have much success. I did find the left side here under "simple pole" https://en....
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### Asymptotic expansion of q-Pochhammer symbol near q = 1

I'd like to understand the asymptotics of the q-Pochhammer symbol $(a;q)_\infty$ as $q \to 1^-$ with $a$ complex, where $$(a;q)_\infty = \prod_{n = 0}^\infty (1- aq^n).$$ More specifically, I'm ...
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### Euler product over a finite subset of composite numbers

The product $$\prod_{p\text{ prime}}\frac{p^n}{p^n-1}=\zeta(n)$$ is well-known. This works over primes, and only over the entire set of them. Is any similar expression known for composites? For a ...
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### Quick questions of infinite series

Are these true: $$1.\sum_{n=1}^{\infty}\frac{1}{n^2+88n}=\frac{1}{88}\sum_{n=1}^{88}\frac{1}{n}$$ $$2.\sum_{n=1}^{\infty}\frac{1}{n^2+4n}=\frac{25}{48}$$ For first one, it's factored. ¿It's not ...
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### On modified Euler product: [closed]

Consider the modified Euler product as follows: $$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$ Here $c$ is a constant My questions are Is there a compact representation for this ...
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### Infinite product representation of $\sqrt{1-z}$

Does there exist an infinite sequence of complex numbers $\{ z_i \}_{i=1,2,\cdots}$ such that  \boxed{ \sqrt{1-z} = \prod_{i=1}^\infty \left( 1 - \frac{z}{z_i} \right) \qquad \textrm{(for } |z|<1)...
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### Why do we say some infinite products “diverge” when the limit is zero, a finite value?

Infinite series (sums) are discussed a lot, but infinite products less so. Despite having tried hard to find reading material on infinite products, I have only found lecture notes and not proper texts....
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### How to represent $e^z -1$ as an entire function

This question was part of my complex analysis assignment and I am not getting ideas on how this question should be attempted. Express $e^z -1$ in an infinite product. I an not able to even start ...
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### Show that This infinite product is entire

This question is from Ponnusamy and silvermann complex analysis and I am making some mistake in this question and unable to find it. Show that $\prod_{n=1}^{\infty} (1-z/n) e^{z/n +5z^2/n^2}$ is ...
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### Showing $\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}$

I want to show \begin{align} \prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})} \end{align} I know one proof via self-conjugation of partition functions with ...
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### Compute the infinite series of the $\displaystyle{\prod_{n=1}^{\infty} \frac{(n+1)^2}{n^2+2n}.}$

Can somebody help me with this. Compute the infinite series of the $\displaystyle{\prod_{n=1}^{\infty} \frac{(n+1)^2}{n^2+2n}.}$ I have no idea how to start with this. This is my first time encounter ...
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### Finding the value of this infinite product

The following question is from my analysis assignment and I was unable to solve it. Find the value of $\prod_{n=1}^{\infty}(1+ 1/n^2)$ . I tried to do some algebraic manipulation but I was not ...
### Is there a closed-form expression for $\prod_{n=1}^{\infty}(1-\frac{x}{n^3})$?
I would like to ask if for $|x|<1$, we can express the product $\prod_{n=1}^{\infty}(1-\frac{x}{n^3})$ as a function $f(x)$. I tried to use Weierstrass factorization theorem, but without much ...