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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} \...
NEON's user avatar
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5 votes
2 answers
142 views

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 ...
Dan's user avatar
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2 votes
1 answer
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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)...
user967210's user avatar
-1 votes
1 answer
61 views

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?
ablobfish's user avatar
6 votes
0 answers
68 views

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 ...
Minh Đức Hoàng's user avatar
1 vote
1 answer
134 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$

I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where $$f(n)=\prod\limits_{k=1}^n \frac{2n}{e}\left(\arcsin\left(\frac{k}{n}\right)-\arcsin\left(\frac{k-1}{n}\right)\right)$$ We have: $f(1)\...
Dan's user avatar
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4 votes
4 answers
111 views

Show that $\int_0^{1/2}\log(n(\arcsin x-\arcsin(x-1/n)))\mathrm dx$ converges to $\frac12-\frac14\log(27/4)$ as $n\to\infty$.

Show that $\lim\limits_{n\to\infty}\int_0^{1/2}\log\left(n\left(\arcsin x-\arcsin\left(x-\frac{1}{n}\right)\right)\right)\mathrm dx=\frac12-\frac14\log\frac{27}{4}$. Here is the graph of $y=\log\left(...
Dan's user avatar
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8 votes
2 answers
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Product of lengths in a disk of area $\frac{\pi}{e}$

A disk of area $\dfrac{\pi}{a}$ is divided into $n$ regions of equal area by line segments from a point on the edge. Here is an example with $n=8$. Let $P(a,n)=\text{product of lengths of the line ...
Dan's user avatar
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0 votes
0 answers
15 views

I want to know various ways to check convergence of infinite product

In complex analysis class I learned some condition that makes infinite product converge. Theorem : If $\sum |a_n|$ converges, then the infinite product $\prod(1+a_n)$ also converges. The proof is ...
SunnyMath's user avatar
  • 175
14 votes
2 answers
428 views

Conjectured connection between $e$ and $\pi$ in a semidisk.

A semidisk with diameter $\dfrac{e}{\pi}n$ is divided into $n$ regions of equal area by line segments from a diameter endpoint. Here is an example with $n=6$. Consider the $n$ arcs between ...
Dan's user avatar
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2 votes
1 answer
35 views

Decomposition of function into products

Given a single variable function $f(x)$, is there a way of decomposing it into the product of a family of function. Something similar to, $$f(x) = \prod_n p^{a_n}_n(x)$$ I am trying to find the ...
PRITIPRIYA DASBEHERA's user avatar
5 votes
1 answer
160 views

Associativity of infinite products

It is well-known that if $\sum_{j=1}^\infty a_j$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\...
Hilbert Jr.'s user avatar
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2 votes
2 answers
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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. ...
wannabemathematician's user avatar
2 votes
0 answers
99 views

About the infinite product $\prod_{n=1}^{\infty}\left(2n-1\right)^{\frac{(-1)^n}{2n-1}}$

I have found a rare infinite product that neither Wolfram Alpha and Mathematica can't evaluate. The product in question: $$\prod_{n=1}^{\infty}\left(2n-1\right)^{\frac{(-1)^n}{2n-1}}=\frac{e^{\pi\...
User's user avatar
  • 359
2 votes
1 answer
408 views

Computation of a finite product

I am preparing for job interviews, and in some old questions list I found this interesting one which I did not manage to crack yet: Compute the following products: $$ P_1 = \Pi_{0< i<j<\infty}...
Lenz's user avatar
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0 answers
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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 ...
Anon's user avatar
  • 33
2 votes
1 answer
90 views

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 ...
Aig's user avatar
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0 votes
0 answers
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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 $H(z-1) = -e^{-\pi(z-1)/2}H(z)$, ...
algebroo's user avatar
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0 votes
3 answers
113 views

Why no one uses the product formula for sine function to calculate $\pi$?

$$\sin(\pi x)=\pi x \prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)$$ $$\pi = \frac{\sin(\pi x)}{x\prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)}$$ Let $x=\frac{1}{2}$ $$\pi = \frac{2}{\prod_{n \ge 1}\...
pie's user avatar
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0 votes
0 answers
112 views

$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Conjecture : $$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$ Where $a_n$ is an integer such that : $$0\leq a_n\leq n$$ Some arguments : ...
Miss and Mister cassoulet char's user avatar
0 votes
0 answers
16 views

Show $\sum_{n=0}^\infty \frac{(cq^n)_{\infty}}{(bq^n)_{\infty}}=\sum_{m=0}^\infty \frac{(c/b)_mb^mq^{nm}}{(q)_m}$

Show $\sum_{n=0}^\infty \frac{(cq^n)_{\infty}}{(bq^n)_{\infty}}=\sum_{m=0}^\infty \frac{(c/b)_mb^mq^{nm}}{(q)_m}$ This was taken from the following equation: $$\frac{(b)_\infty}{(c)_\infty} \sum_{n=0}^...
Nishi's user avatar
  • 31
0 votes
2 answers
71 views

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 ...
HighAsAKiteOnMath's user avatar
1 vote
0 answers
63 views

Completion of an infinite tensor product of Banach spaces: can an infinite product converge to 0?

Given a Banach vector space $X$, I need to consider an infinite tensor product of it. The simplest construction I found, in Guichardet, is by considering the inductive limit of the direct system $(X^{\...
Charlie's user avatar
  • 11
12 votes
2 answers
312 views

Prove that Wallis' product and Euler's formula directly imply that $(-1/2)!=\sqrt{\pi}$

(This occured to me recently, and I was pretty sure that it was true, so I was pleased that it really was. This has almost certainly been published many times before, but I didn't see it in either of ...
marty cohen's user avatar
1 vote
1 answer
41 views

Finding domain of convergence of an infinite product

I would like to find the domain of convergence of $\displaystyle{\prod_{n=1}^\infty \left(1+\left(1+\frac{1}{n}\right)^{n^2}z^n\right) }$, where $z$ is a complex number. From my understanding, an ...
obitobi_tobias's user avatar
0 votes
1 answer
121 views

Distributive property proof.

I was reading this question here: Distributivity of categorical product and sum but I could not understand the statement of the OP that said "If $\textbf{C}$ is $\textbf{Set}$ or $\textbf{Top}$ I'...
Hope's user avatar
  • 103
3 votes
1 answer
114 views

Does $\prod\limits_{n=1}^{\infty}\int\limits_{-1}^{1}\left|\frac{\lfloor t^{1-n}\rfloor}{\lfloor t^{-n}\rfloor}\right|dt$ converge to $1$?

This is the first $40$ partial products on Desmos (Desmos gave me "undefined" for everything higher): Looking at this, it doesn't appear to converge but then, while attempting to get Desmos ...
Dylan Levine's user avatar
  • 1,666
2 votes
2 answers
218 views

Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$

Question statement Evaluate the infinite product $$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$ My try Because of the square of $\displaystyle{x}$ , we can consider $...
Martin.s's user avatar
  • 4,313
4 votes
1 answer
146 views

Cantor's representation of a real as infinite product

Studing representation of real number, i have found (2.13) in The Real Numbers - A Survey of Constructions by Ittay Weiss a Cantor theorem: Every real number $a > 1$ can be written uniquely as $a =\...
user791759's user avatar
-1 votes
1 answer
76 views

Solve an infinite product. [closed]

$a_n = (1+ z ^{2^{n-1}})$, where $n$ is a natural number. Prove that $\prod_{n=1}^{\infty} a_n = \frac{1}{1-z}$, where $z\in \mathbf{C} \text{ and } |z| < 1$ Please help me solving this question. I ...
Guru P L's user avatar
3 votes
0 answers
69 views

$\left(-\frac{1}{2}\right)!$ from the Hypersphere

I was deriving the formula for volume of a n- dimensional Hypersphere when I came across something interesting. It seems like we can define $(0.5)!$ without resorting to the Gamma function. Define: $$...
Tanmay Gupta's user avatar
1 vote
0 answers
52 views

$q$-Pochhammer at root of unity

Are there any identities, papers/studies, posts, etc that go over $$(\ln\zeta_n^k;q)_{\infty} = \prod_{m=0}^{\infty}(1-\frac{2\pi i k q^m}{n})$$ which is sometimes called the $q$-Pochhammer or quantum ...
Mako's user avatar
  • 576
6 votes
0 answers
246 views

Show that $e^z-1=z e^{\frac{z}{2}} \prod_{n=1}^{\infty}\left(1+\frac{z^2}{4 n^2 \pi^2}\right)$ using Residue Theorem

I was studying previous complex analysis exams and I came across the following question: Use a result obtained from the residue theorem to justify that following infinite product is representation is ...
Tropax's user avatar
  • 373
0 votes
0 answers
47 views

Evaluate $\prod_{n=2}^\infty(2-2^{1/n})$ [duplicate]

I tried telescoping the product but coudn't find a way to do so, any leads on how to evaluate this limit will help. $$ \lim_{n\to\infty}(2-2^{1/2})(2-2^{1/3})\cdots(2-2^{1/n}) $$
randoLorries's user avatar
0 votes
0 answers
59 views

Infinite products and Maclaurin series.

Studying about the way Euler answered the Basel Problem, I was interested about functions defined by infinite products. So I developed a way to expand a product by the "general" form of the ...
Tio Zuca's user avatar
  • 340
2 votes
0 answers
62 views

Convergence of infinite products of formal power series

Setting Let $R$ be a domain. The ring $R[[X]]$ of formal power series is a complete ultrametric space, see this Wikipedia article. According to the same source, "the philosophy of formal power ...
azimut's user avatar
  • 22.8k
2 votes
1 answer
243 views

Equivalence statement for existence product of modules

The following exercise is taken from T.S. Blyth book, Module Theory, chapter 6, exercise 9: Something does not add up, especially with using (1). It seems that condition (1) is not necessary or ...
User666x's user avatar
  • 846
0 votes
1 answer
90 views

Can the infinite product of the powers of a transcendental number ever be transcendental?

I plugged the following product into a calculator: $$\prod_{n=1}^\infty e^{\frac{1}{n^2}}$$ and got a result of roughly 5.1806683. I would like to say that this result is transcendental, as its only ...
Alexandra's user avatar
  • 453
3 votes
0 answers
99 views

$\epsilon$-proof for convergent infinite product [duplicate]

Setting There is the following statement about infinite products: If $\prod_{n=0}^\infty a_n$ is convergent (which by convention means that the limit is not zero), then $\lim_{n\to\infty} a_n = 1$. ...
azimut's user avatar
  • 22.8k
0 votes
1 answer
107 views

Problem in understanding Blaschke product.

Blaschke product is defined in the following way $:$ $$B(z) = \prod\limits_{n = 1}^{\infty} \frac {|z_n|} {z_n} \frac {z_n - z} {1 - \overline z_n z},\ z_n \in \mathbb D \setminus \{\textbf 0\}\ \...
Akiro Kurosawa's user avatar
5 votes
3 answers
123 views

A good upper bound for the infinite product $\prod_{i=0}^\infty (1+p/2^i)$

What is a good/tight upper bound for the infinite product $$\prod_{i=0}^\infty \left(1+\frac{p}{2^i}\right)$$ as $p\to \infty$? Indeed, it's bounded above by $e^{2p}$ so at least its finite, but that ...
Andrew Yuan's user avatar
  • 2,932
6 votes
2 answers
212 views

I have multiple questions regarding one problem: For a positive real $a$ what's the value of $\sqrt{a\sqrt{a\sqrt{a...}}}$

I'm reading a book, the book is aimed at high schoolers so it is perhaps not the most rigorous book. The intended solutions the book provides for the problem are the following: First assume the ...
zlaaemi's user avatar
  • 1,077
1 vote
2 answers
138 views

Infinite Product Problem

I need some help with this infinite product (or rather limit of partial products) which really gives me a headache. It goes like that: $$\lim_{n \to \infty} n\prod_{m=1}^{n} \biggl(1 - \frac{1}{m} + \...
user1265841's user avatar
1 vote
1 answer
74 views

Can infinite limit on products of N terms ignore $\mathcal{O}(N^{-2})$ in each term?

Is the following true in general? $$\lim_{N\rightarrow \infty}\prod_{n=1}^N\left(a_n+b_n\left(\frac 1 N\right) + \mathcal{O}\left(\left(\frac 1 N\right)^2\right)\right) = \lim_{N\rightarrow \infty}\...
372191's user avatar
  • 13
29 votes
2 answers
6k views

Find a simple proof that π is irrational

I know there are many questions on the site about finding a proof that π is irrational, but I'm posting the question separately to discuss a particular proof further We know that the Wallis Product is ...
زكريا حسناوي's user avatar
0 votes
0 answers
43 views

Conditions equivalent to uniform convergence of infinite product of functions

Let $(f_n(z))$ be sequence of complex functions. By definition, infinite product $\prod f_n$ converges uniformly if there exists $N\in \mathbb{N}$ such that $F_N(z):=\lim_{m\to \infty} \prod_{n=N}^{m} ...
Rain's user avatar
  • 117
1 vote
1 answer
90 views

What is the value of this infinite product?

I was thinking a way to evaluate the product $$\prod_{n=1}^\infty \frac{1}{1-2^{-n}}=\prod_{n=1}^\infty \left(1+2^{-n}+2^{-2n}+\dots \right)$$ I just thought about the function $f(x)=\prod_{n=0}^\...
Tio Zuca's user avatar
  • 340
12 votes
0 answers
383 views

Infinite sum of Gamma random variables with same shape parameter but different rate parameter

Question Let $Z_i\sim\chi_{(1)}^2\sim\Gamma(\frac{1}{2},\frac{1}{2})$ be i.i.d. chi-square random variables. We define: $$ W_n =\Bigl[\sum_{i = 1}^{n-1}\frac{Z_i}{2^{i}}\Bigr] + \frac{Z_n}{2^{n-1}} \...
MathRevenge's user avatar
1 vote
0 answers
58 views

Is this statement about functions expressed as infinite products true?

Let $f$ be a function of $x$ that respects these conditions: $f(a_n)=0,f(0)=1, n \in [1,\infty) $. From this, can we express the function $f$ as an infinite product? If so, is this statement true? $$f(...
Tio Zuca's user avatar
  • 340
2 votes
0 answers
192 views

Find the value of $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\cdots \infty}}}$? [duplicate]

Greetings with utmost respect, everyone! Today, I found a fascinating math question online. I seem to be stuck while solving it, however. I did not find any relevant solution to the problem yet. ...
Rohan Bari's user avatar

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