Questions tagged [infinite-product]

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

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46 views

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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1answer
68 views

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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1answer
30 views

Translation symmetry by Gaussian integers for this infinite product is impossible by Liouville. What takes its place?

On $\Bbb{R}$, we can construct the convergent infinite product $$F_{\Bbb{Z}}(x) := x \prod_{n \geq 1} \prod_{|k| = n} \left(1 - \frac{x}{k} \right) = x \prod_{n \geq 1} \left(1 - \frac{x^2}{n^2} \...
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0answers
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Conditional convergence of an infinite series of logarithms

Let $\rho_n$ be a sequence of positive real numbers converging to $0$. Suppose that the following series is convergent $$ \sum\log(1 + \rho_n(2\cos(\theta_n) + \rho_n)). $$ We may assume that the ...
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2answers
42 views

Showing $\sinh z=z\prod_{n=1}^\infty\left(1+\frac{z^2}{n^2\pi^2}\right)$, given $\sin z=z\prod_{n=1}^{ \infty}\left(1 - \frac{z^2}{n^2\pi^2}\right)$

How to show that $$\sinh z=z\prod_{n=1}^\infty\left(1+\frac{z^2}{n^2\pi^2}\right)$$ using the well-know representation of sine as infinite product that is $$\sin z=z\prod_{n=1}^{ \infty}\left(1 - \...
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2answers
52 views

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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2answers
50 views

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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3answers
70 views

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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1answer
164 views

Closed form of the sum $\sum _{n=1}^{\infty }\:\left(\frac{1}{x^2-n^2\pi ^2}\right)$

Came across this in an old textbook and I'm struggling to simplify this in any way. I tried to integrate it and write it as a product: $\int y\:=\:\frac{1}{2x}\left(\log\prod \:_{n=1}^{\infty }\left(x^...
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Proving that odd partitions and distinct partitions are equal

I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998). Corollary 1.2 is a standard result that shows that the number of partitions of $...
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1answer
24 views

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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1answer
63 views

Finding the closed form of $\prod _{n=0}^{\infty }\left(1\:-\:x^{2^n}+x^{2^{n+1}}\right)$ [closed]

As far as I can tell this expression doesn't telescope. How would one go about this?
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2answers
41 views

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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1answer
55 views

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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0answers
63 views

What does $\left[\pi \cdot \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \ldots \ldots \infty\right]$ equals?

I recently came across this interesting question Find the value of $\left[\pi(\frac{2}{1}) (\frac{2}{3} )(\frac{4}{3}) (\frac{4}{5}) (\frac{6}{5})(\frac{6}{7})(\frac{8}{7})(\frac{8}{9}) \cdots (\...
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1answer
52 views

Limit of a continued product $\prod_{r=3}^n \frac{(r^3+3r)^2}{r^6-64} $ as $n \to \infty$ [closed]

Evaluate $\lim_{n\to \infty }\prod_{r=3}^n \frac{(r^3+3r)^2}{r^6-64} $ I don't have any ideas on how to approach the problem. Hints or solutions are appreciated.
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49 views

Is this infinite product in $L^1(\mathbb{R}, m)$?

$$ f(x)=\frac{\sin(x/2)}{x/2}\prod_{k=1}^{\infty}\cos\left(\frac{x}{3^k}\right), \hspace{1em} x\in\mathbb{R}\backslash\{0\} $$ and $f(0)=1$ to achieve continuity. $m$ denotes the Lebesgue measure. If ...
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2answers
67 views

How to show that $\prod_{p\text{ prime}} \frac{p}{p-1}$ diverges

I was just reading a Wikipedia article regarding the existence of infinitely many primes in certain infinite arithmetic progressions, and I read something interesting- that Euler had once discovered (...
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2answers
43 views

$\prod_{k=1}^{\infty}(1-\alpha_k)>0\iff\sum_{k=1}^{\infty}\alpha_k<\infty\;\;\;\ \alpha_k\in(0,1)$ Proof verification

I would like to ensure that my logic works here for proving the following: $$\prod_{k=1}^{\infty}(1-\alpha_k)>0\iff\sum_{k=1}^{\infty}\alpha_k<\infty\;\;\;\;\;\;, \alpha_k\in(0,1)$$ So, for $\...
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3answers
202 views

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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1answer
62 views

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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1answer
42 views

$\prod_{n=1}^N (1+f_n(x))$ converges uniformly to a function on $A$ as $N\to\infty$.

I am reading "Calculus vol.2" (in Japanese) by Shizuo Miyajima. There is the following theorem in this book: Theorem 8.2 Let $f_1, f_2, \dots$ be functions on a set $A$. Let $M_1, M_2, \...
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1answer
75 views

What subset of the reals have unique prime factorizations if you allow rational exponents?

By the fundamental theorem of arithmetic we know that all positive integers have a unique prime factorization. So if $n$ is some positive integer, then $$n = \prod_{i\in\mathbb{N}}{p_i^{e_i}}$$ where $...
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1answer
111 views

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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1answer
60 views

Values of the infinite product $\prod_n\frac{(n+1)x}{1+nx}$

I am trying to compute the inverval of convergence and the explicit value of the infinite series $$\prod_{n=1}^{\infty}\left(\dfrac{(n+1)x}{1+nx}\right).$$ I believe the interval of convergence is $(-...
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1answer
54 views

Proof Leibniz pi series using Wallis product or viceversa

Hi I am trying to get a smart way to connect this two well known expressions for $\pi$: Leibniz serie: $$\frac{\pi}{4}=\sum_{n=1}^{\infty}{\frac{(-1)^{n+1}}{2n-1}}$$ Wallis product: $$\frac{\pi}{2}=\...
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2answers
42 views

Closed form of infinte product proof: $\prod_{i=1}^{n}(1 + x^{2^{i-1}}) = \frac{1-x^{2^n}}{1-x}$

I was working through problems in "An Introduction to Mathematical Reasoning" and am stumped on an induction proof in Problems I: Question 18, which is: $$\prod_{i=1}^{n}(1 + x^{2^{i-1}}) = \...
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0answers
47 views

An infinite product involving triplets of consecutive prime numbers

Let $p_n$ be the $n$th prime, $n\ge 1$. I was wondering if the following product converges $$P=\prod _{k=1}^{\infty} \frac{p_{k+1}}{\sqrt{p_k p_{k+2}}}$$ I have computed some numeric examples and the ...
7
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1answer
126 views

An infinite product for $\frac{\pi}{2}$

Please help prove $$ \begin{align} \frac{\pi}{2}&=\left(\frac{1}{2}\right)^{2/1}\left(\frac{2^{2}}{1^{1}}\right)^{4/(1\cdot 3)}\left(\frac{1}{4}\right)^{2/3}\left(\frac{2^{2}\cdot4^{4}}{1^{1}\...
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1answer
98 views

Need help on understanding a proof of the formula for calculating the $\zeta(2n)$

In this thread, Jack D'Aurizio provided a succinct proof for the formula of calculating the values of the Zeta function $\zeta(2n)$ $$ \coth z-\frac{1}{z} = \sum_{n\geq 1}\frac{4^n\,B_{2n}}{(2n)!}z^{...
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1answer
68 views

What mathematical operation replaces series of multiplication

Technical details because I might misunderstand my problem... I have made an algorithm that calculates, what camera view frustum is needed to view a mesh so that it would be fully visible in the ...
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1answer
75 views

Difference between the definition of the dot product in infinite and finite dimensions

I know that in $\mathbf{\mathbb{R}}^n$ the definition of the dot (or scalar) product is the following: $x.y=x^{\mathrm{T}}y$, with ''T" denoting the transpose of the vector x. How does this ...
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0answers
51 views

Has this equation $\prod_{n=1}^{\infty}(1-\frac{z}{n^a})e^{\frac{z}{n^a}+\frac{z^2}{2n^{2a}}}=a.$ solutions?

With regard to the following equation: $$\prod_{n=1}^{\infty}(1-\frac{z}{n^a})e^{\frac{z}{n^a}+\frac{z^2}{2n^{2a}}}=a,$$ I am trying to answer the following questions: for $a=\frac{\pi}{7}$, has the ...
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2answers
96 views

Find what a Infinite product number approaches

Find what the following number approaches: $$\frac{6}{5}\times\frac{26}{25}\times\frac{626}{625}\times\frac{390626}{390625}\times...$$ What I found is that each fraction can be simplified as:$$1+\...
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3answers
360 views

Proving $\prod_{n=1}^{\infty}\left(1+\frac{1}{n^{3}}\right)=\frac{1}{\pi}\cosh\frac{\pi\sqrt{3}}{2}$

First I rewrite $\prod_{n=1}^{\infty}\left(1+\frac{1}{n^{3}}\right)$ as $\prod_{n=1}^{\infty}\left(\frac{1+n^{3}}{n^{3}}\right)$, then by factor out polynomial I get $\prod_{n=1}^{\infty}\left(\frac{(...
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0answers
34 views

How can we extend the notion of unique factorization monoid to arbitrary products?

We have a precise definition of unique factorization monoid that concerns products of finitely many elements. How can we extend it to arbitrary products, as the ones that we find in McKenzie's article ...
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1answer
33 views

compute $\prod_{n=2}^{N} \frac{n(n+1) +1}{ n(n-1)+1}$ .

I am not able to compute this infinite product and so I am asking for help here. Compute the product $\prod_{n=2}^{N} \frac{n(n+1) +1}{ n(n-1)+1}$ . I tried by factoring $\frac{n(n+1) +1}{ n(n-1)+1}$...
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5answers
370 views

How to prove $ \prod_{n=1}^{\infty} \left(1+\frac{2}{n}\right)^{(-1)^{n+1}n} \,= \frac{\pi}{2e}$

How can I prove that $$ \prod_{n=1}^{\infty} \left(1+\frac{2}{n}\right)^{\large{(-1)^{n+1}n}} \,= \frac{\pi}{2e}$$ The result is given here (result 48). The source: Prudnikov et al. 1986, p. 757 is ...
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0answers
118 views

A curious infinite product of factorials

I found the following infinite product of factorials without proof: $$\small\prod_{n=1}^\infty\frac{{(2 n)!}^{20}\,{(8 n)!}^{32}\,{(32 n)!}^2}{{n!}^4\,{(4 n)!}^{37}\,{(16 n)!}^{13}}=\\\frac{\sin ^{14}\...
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2answers
42 views

Convergence/Divergence proof for Infinite products

Exercise 2.4.10 is about infinite products. I'd like someone to verify, if my convergence/divergence proof is technically correct and rigorous. A close relative of the infinite series is the infinite ...
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0answers
46 views

Convergence of a sequence of products

$\{c_{jn}\}$ be complex numbers for $j,n\in\mathbb{N}$ s.t. $c_{jn}=0$ if $j>n$. If $m_n=\text{max}_{1\le j\le n}\left|c_{jn}\right|$ then one has $\lim_{n\rightarrow\infty}m_{n}=0$. If $s_n=\...
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0answers
124 views

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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1answer
77 views

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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0answers
42 views

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 ...
2
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2answers
110 views

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 ...
3
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1answer
63 views

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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2answers
59 views

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 ...
0
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1answer
32 views

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 ...
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1answer
62 views

Construct an entire function whose only zeroes are following

This question is from Ponnusamy and Silvermann Complex Variables pg 436, subsection wietestrauss product theorem Question: Construct an entire function whose only zeroes are z=ln n. Weierstrauss ...
6
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2answers
107 views

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 ...

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