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Questions tagged [infinite-product]

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

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Approximation of $\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)$

After this post, I started wondering about possible approximations of the infinite product $$A_p=\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)\tag 1$$ where $p$ is an integer. As far ...
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How to work out this finite product?

$$\prod_{k=1}^{4n-2}\left[\sin\left(a+\frac{k\pi}{4n-2}\right)+\cos\left(a+\frac{k\pi}{4n-2}\right)\right]^{(-1)^k}\tag1$$ Suppose $a\ge0 $ and $n\ge1$ How to verify that $$(1)=(-1)^n\left[\frac{1-\...
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2answers
99 views

Simplifying $\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$

I am looking to simplify the following, without the use of capital Pi notation: $$\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$$ Which is meant to produce the sequence: $\left[1,\ \frac{1}{2},\ \...
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23 views

difficulty evaluating solving this gamma function value

I am aiming to solve the following $$\prod_{n=1}^{\infty} \left(1-\frac{1}{(2n)^3} \right) $$ Note its similarity to $$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3} \right)=\frac{\cosh(\frac{\pi}{2}\...
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2answers
47 views

Product of $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ [duplicate]

I know that answer to $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ = $(1+q)(1+q^{2})(1+q^{4})...$ = $\frac{1}{1-q}$. And i need to prove this equality using generating functions. Any hints?
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64 views

Show $\cosh(z) - \cos(z) = z^2\prod_{n = 1}^{\infty}\left(1 + \frac{z^4}{4\pi^4n^4} \right)$

I would like to show that $\cosh(z) - \cos(z) = z^2\prod_{n = 1}^{\infty}\left(1 + \frac{z^4}{4\pi^4n^4} \right)$. Firstly, I have found that solutions to the equation $\cosh(z) - \cos(z) = 0$ are of ...
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1answer
48 views

How can i prove the following Equality? involving these infinite products

$$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right)= \frac{\cosh(\frac{\pi}{2}\sqrt3)}{3\pi} $$ This can be found here (http://mathworld.wolfram.com/InfiniteProduct.html) Line 22 It is known that $$...
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179 views

Compute $ \xi_p= \prod_{n=1}^{\infty} (1+\frac {1}{n^p})$

The main question I want to ask is inspired from this question Find the value of $$\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)$$ Now, I have solved this question easily using product ...
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How can i prove the following Infinite Product?

I understand how to derive the Euler Product over primes for the Zeta function using the sieving method. However, upon reading more it said the following.. $$\prod_{P Prime}^{\infty}(1-p^{-s})^{-1} = ...
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Understanding infinite product of $sin(\pi z)$

The infinite product of $sin(\pi z)$ is said to be... $\sin \pi z=\pi z\prod _{{n\neq 0}}\left(1-{\frac {z}{n}}\right)e^{{z/n}}=\pi z\prod _{{n=1}}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)$ ...
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1answer
49 views

Understanding the roots in the Weierstrass factorization theorem

To construct the Weirstrass product, we first start with something of the form $$\prod_{k=1}^\infty(z-a_k)$$ The next thing we do is include exponential factors out front (which obviously don't ...
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1answer
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How can I express $3 \cdot 7 \cdot 11 \cdots (4n+3)$ in terms of factorial?

This is the work I have done so far: $\prod_{k=0}^n(4k+3) = \frac{(4n)!}{2^n(2n)!}\cdot\prod_{k=0}^n\frac{1}{4k+1}$. I would really appreciate a clever trick how to reduce the latter product that ...
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2answers
66 views

How to use Taylor Series to find infinite power of $\sin x$

I am told that the Taylor Series for $\sin(x)$ is... $$p(x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!}$$ I know that the product for a finite polynomial has the form... $$ p(x) = ...
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2answers
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Evaluating: $\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $

How would we evaluate: $$\lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) $$ Is it possible to evaluate this manually? Or do we have to make a program to get an ...
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1answer
80 views

Understanding the elementary factors in Weierstrass factorization theorem

An infinite product such as $\,\prod _{n}(z-c_{n})$ cannot converge. In order for it to converge, each factor $(z-c_{n})$ must approach 1 as $n\to \infty$. So it stands to reason that one should seek ...
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1answer
41 views

Understanding infinite products convergence

I am reading about a convergence problem with infinite products and I am told: Any finite sequence $\{c_{n}\}$ in the complex plane has an associated polynomial p(z) that has zeroes precisely at ...
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1answer
47 views

How to prove that $\frac{1}{1-z}=\prod_{n=0}^\infty (1+z^{2^n})$?

A simple pole can be written as $\displaystyle{\frac{c}{c-z}=\prod_{n=1}^\infty e^{\frac{1}{n}\left(\frac{z}{c}\right)^n}}$. How does one show that when $c=1$, $\displaystyle{\frac{1}{1-z}=\prod_{n=0}^...
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2answers
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Ratio of $\frac{\zeta(2n)}{\zeta(n)}$ from infinite product involving primes

Given that $$\zeta(n)=\sum_{k=1}^\infty \frac{1}{k^n}=\prod_{k=1}^\infty \frac{1}{1-\frac{1}{(p_k)^n}}\tag{1}$$ where $n>1$ and $p_k$ is the $k^{th}$ prime. Proof of the Euler product formula for ...
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1answer
303 views

Proof of identity for $\pi$: $\frac{\pi}{3} = \frac{2}{\sqrt{2+\sqrt{3}}}\frac{2}{\sqrt{2+\sqrt{2+\sqrt{3}}}}\cdots$

While browsing the internet today, I came across the following picture: (full image can be found here - credit to Цогтгэрэл Гантөмөр) Now, it would naturally seem we can extend this to an infinite ...
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85 views

closed form of $\prod_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{(-1)^{n-1}n}$

I am looking for the closed form of this product. $$\prod_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{(-1)^{n-1}n}$$ I have sees it somewhere before but I can't remember it closed form. I remember the ...
11
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1answer
253 views

product= $\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\left(\frac{11^{11}3^3}{13^{13}}\right)^{1/20}\sqrt{\frac{3}{7^{7/6}\pi}\sqrt{\frac2\pi}}$

$\mathrm G$ is Catalan's constant. I recently found the product $$ \alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
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0answers
39 views

Defining the notion of a product over an uncountable set

Background: So a lurking thought in the back of my mind for a month or so has been the notion of a sum over an uncountable set, say $$\sum_{a \in I} a$$ for some $I$ whose cardinality is greater ...
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30 views

Prove that $\sin(\pi z)$ can be written as infinite product [duplicate]

Prove that \begin{align} \sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2}\right) \, \, \, \, \forall \, z \in \mathbb{C} \end{align} The hint I had it's to use the Fourier series, ...
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1answer
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Find the value of $\prod_{k=1}^{\infty} \big(1+\frac{1}{k^s}\big)$

I am attempting to find values to the family of products given by $$p(s)=\prod_{k=1}^{\infty} \Bigg(1+\frac{1}{k^s}\Bigg)$$ where $s\in \mathbb{C}$ and the real part of $s$ is greater than $1$. In ...
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1answer
38 views

Does the following Infinite product converge to anything?

The infinite product in question is $$\prod_{n=1}^{\infty}(1-\frac{x}{n\pi})$$ I see that this product is similar to that of $$\frac{sin(x)}{x}= \prod_{n=1}^{\infty}(1-\frac{x^2}{n^2\pi^2})$$ ...
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2answers
70 views

General closed form for $L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx$ when $\phi\in(0,\pi)$?

I would like to know if there is a general closed form for $$L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx,\qquad \phi \in(0,\pi)$$ Context: (below are also the extent of my search for a closed form....
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1answer
59 views

Convergence of $\sum\limits_{n=0}^{\infty} (1-|a_n|)$

So I must prove that if $(a_n)$ is a sequence of points in $\mathbb{C}$ with $0< |a_n| < 1 \; \forall n \in \mathbb{N}$ and verifying that $|b| \leq \prod\limits_{n=1}^{\infty} |a_n|$ with $0&...
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1answer
13 views

Is the following statement about matrix norms of matrix products correct?

Let $A$ be a real-valued, square matrix and define its 2-norm as: $$||A||_2 = \sqrt{\max_i\lambda_i(AA^T)}$$ where $\lambda_i(AA^T)$ denotes the $i^{th}$ eigenvalue of the product $AA^T$. Now ...
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1answer
44 views

Simplifying a product of a series

I tried simplifying the product $$\prod_{k=1}^{\infty}\left[1-x^k\right]$$ by factoring it into $$\prod_{k=1}^{\infty}\left[\left(1-x\right)\sum_{i=0}^{k-1}x^i\right].$$ I am not very experienced in ...
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34 views

Infinite products of Laurent series

I am trying to find an expression for the coefficients of a Laurent series which is itself an infinite product of Laurent series: $f(z) = \sum_{u=-\infty}^{\infty} f_{u}z^{u} = \prod_{i=0}^{\infty} \...
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24 views

Do the Airy Functions have a product representation?

I was wondering if the Airy Functions $$Ai(z)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(zt+\frac{t^3}{3})}dt$$ $$Bi(z)=\frac{1}{\pi}\int_0^\infty e^{-\frac{t^3}{3}+tz}+\sin\left(\frac{t^3}{3}+zt\right)...
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1answer
21 views

Why can('t) I map this closed subset of a product of compact sets to a non-compact set?

Let $S$ be a countable set of real numbers that is bounded but has neither a maximum nor a minimum. Next we create the product topology $[0, 1]^S$ (using the usual topology on $[0, 1]$). This should ...
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2answers
145 views

How to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$?

I need to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$ but wolframalpha's brain is too narrow to contain the result, and I don't know any software and programming to find ...
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0answers
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Methods for assigning values to infinite constructs (ex. sums, products)

(This question is more to get an overhead view of the topic rather than a well defined answer) How can one assign values to infinite constructs AND justify them. What do i mean with infinite ...
4
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1answer
97 views

Evaluate $\frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}\cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots$

Problem Evaluate the infinite product $$\frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}\cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots$$ Attempt For convenience,let's rewrite the limit. ...
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2answers
238 views

When does the tensor product distribute over an infinite direct product?

It is well known that the tensor product of $R$-modules over some ring $R$ does not, in general, distribute over infinite direct products, an obvious example being $\mathbb Z_p \otimes_\mathbb Z \...
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2answers
47 views

Demonstrating an equivalent formula for $\sin(x)/x$ using power series

Would appreciate some ideas for the following: "Prove that $\frac{\sin{x}}{x}=\prod_{n=1}^{\infty}\cos{\frac{x}{2^n}}$ using power series." I'm aware this identity can be shown using trig identities ...
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1answer
32 views

Verifying that $ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \frac{1}{(1-q^{2j-1})(1-q^{2j})}$

On page 165 of Chapter 13, how was the equality made from line 1 to line 2? https://archive.org/details/NumberTheory_862/page/n173 Namely, how $$ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\...
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1answer
738 views

Integral $\int_0^1 \frac{dx}{\prod_{n=1}^\infty (1+x^n)}$

The following integral appeared this summer on AoPS. However it received no answer until today. $$I=\lim_{n\to \infty } \int_0^1\frac{dx}{(1+x)(1+x^2)\dots(1+x^n)}=\int_0^1 \frac{dx}{\prod_{n=1}^\...
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2answers
45 views

How can I show that this infinite product is nonzero?

How would you show that $\prod_{k=1}^\infty \cos( 2 \pi/3^k)$ is nonzero? Wolfram approximates it as about $-0.37$, and I have a guess that $$ \Big \vert \prod_{k=1}^\infty \cos( 2 \pi/3^k) \Big \...
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1answer
51 views

Does the infinite product of the reciprocals of a decreasing (or increasing) function equal zero? [closed]

I've made a short document explaining what I've just claimed. I'd like to know if the criteria for the infinite product to be zero is enough to hold for all decreasing and increasing functions. ...
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0answers
189 views

If a product of polynomials converges, does some product of their zeros also converge?

Suppose $\{p_k\}$ is a sequence of polynomials with $p_k(0)=1$. Let $a_1,a_2,\ldots$ be an enumeration of all of the zeros of the $p_k$. Suppose that $$\prod_{k=1}^\infty p_k(z)$$ converges ...
7
votes
3answers
114 views

Prove that $\prod_{i=1}^n (1+x_i/n) \sim \exp (\sum_{i=1}^n x_i/n)$ as $n\rightarrow\infty$?

Let $x_1,x_2,\dots$ be an infinite sequence of real numbers. Assume that they are bounded, $|x_i| \le C < \infty$ for all $i$ for some $C$. Is it true that, for any such sequence $$\lim_{n \...
10
votes
2answers
146 views

Show that $\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$ [duplicate]

$$\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$$ This monstrous expression is from Tolaso Network (tolaso.com.gr). ...
4
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1answer
93 views

Is this zeta-type function meromorphic?

In An older question I asked : ( See A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH) ) —— Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ ...
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0answers
40 views

Need someone to show me how the Zeta function is equal to Euler's product formula?

$\zeta(x)$=$1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+\frac{1}{6^x}+\cdots$Euler's product formula states, for all $x$ greater than $1$ we have:$\zeta(x)$=$1\over 1-\frac{1}{2^x}$$\...
1
vote
3answers
74 views

Solution to infinite product $\prod_{p-primes}^{\infty} \frac{p}{p-1}$

I want to find the $\prod_{p-primes}^{\infty} \frac{p}{p-1}$. This question stems from a question from amc 12a 2018. It goes as follows: Let $A$ be the set of positive integers that have no prime ...
2
votes
0answers
20 views

What is the terminology for a subset of a product of sets that is the product of its cross-sections?

Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind ...
0
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1answer
34 views

Convergence to zero of a specific infinite product

Maybe less ambitious but I would be happy to prove the following. I have tried but without success. Thanks for any help
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0answers
40 views

Particular infinite product convergence

My question is a little bit technical but if someone has a clue... I have the following infinite product $$ P = \prod_{n=1}^\infty( 1 - q^n(n)) $$ where $q(n)$ is an increasing sequence whose limit ...