Questions tagged [infinite-product]
For questions on infinite products: convergence, computation, etc...
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An entire function that only has simple zeroes at the positive squares
By the Weierstrass Factorization Theorem, every entire function f can be represented as a product involving its zeroes.
Moreover, if $\{ a_n \}$ is the sequence of zeroes of $f$ then $\displaystyle f(...
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Asymptotics for $ f(n) = \prod_{k=2}^{n} \sqrt[k-1]{k}$?
I was thinking about this function
$$f(n) = \prod_{k=2}^{n} \sqrt[k-1]{k}$$
Maybe use it to do number theory or so.
But then I started to wonder about the asymptotics of $f(n)$.
I first assumed $g(n) =...
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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 ...
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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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Showing that the limit of an infinite product equals a sine function times a t term
I need help with a textbook exercise (Gamelin's Complex Analysis, Ch.8 Section 3 Problem 14). This exercise requires me to show that $$\lim_{n\to\infty} \prod_{-m\leq k \leq tm} (1+\frac{z}{k}) =\...
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The product of a sequence of certain product of primes
Let's say I want to calculate:
$$\lim_{n \to \infty} \frac{\prod_{k=2}^{n}{p_{k}*p_{k}}}{\prod_{k=2}^{n}({p_{k}-2)*p_{k+1}}}$$
where $p_{k}$ is the $k^{th}$ prime.
Basically this is $(3*3)/(1*5) * (5*...
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The limit of the product of $\frac{n}{n-k}$ where $k$ is constant and $n\rightarrow\infty$
How do I calculate the limit of products such as:
$$\lim_{n \to \infty} \frac{\prod_{x=k+1}^{n}x}{(n+1)\prod_{x=k+1}^{n}(x-k)}$$
First of all, I am assuming this is independent of k, since k is a ...
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Convergence of infinite product and its limit
I wanted to find $\prod_{n=2}^{\infty}(1+\frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+...)$ and ended up simplifying it as
$\prod_{n=2}^{\infty} \frac{n^2}{n^2-1}$. Now the partial product is $\frac{2n}{...
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Computing infinite product of differences
I'm reading Quantitative Portfolio Management by Isichenko, and one of the interview questions shared in the book is as follows:
Compute the products:
a. $\prod_{0 < i < j < \infty} (i^{1/i} ...
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What is the probability that $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...>e$, where each $u$ is a uniformly random real number in $(0,1)$?
What is the probability that $\prod\limits_{k=1}^\infty \left(1+\prod\limits_{i=1}^k u_i\right)>e$, where the $u$'s are i.i.d. $\text{Uniform}(0,1)$-variables ?
The product, $\prod\limits_{k=1}^\...
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What is the expectation of $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...$, where each $u$ is a uniformly random real number in $(0,1)$?
What is the expectation of $\prod\limits_{k=1}^\infty \left(1+\prod\limits_{i=1}^k u_i\right)$, where the $u$'s are i.i.d. $\text{Uniform}(0,1)$-variables ?
My first thought was to try replacing $u_i$...
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Seeking a proven explicit or closed form upper bound on the infinite product of the Zeta function over the natural numbers, 2 to infinity
I'm interested in upper bounds on the product $\prod_{n=2}^\infty \zeta(n)$
The following post was very helpful and from the answers I see I'll need to study up on partitions to learn more about this ...
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Left Hand Limit of this continous product [closed]
Let $$\lim_{x \to 1^-} \prod_{n=0}^{\infty} \left[\frac{1+x^{n+1}}{1+x^n}\right]^{x^n}=L.$$
Find $\left\lceil 1/L \right\rceil$.
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Prove that $\prod _{n=1}^{\infty } \left(1-\frac{x}{\pi n}\right) \left(\frac{x}{\pi n}+1\right)=\sin(x)/x$ [duplicate]
Would anyone know of a reference where the following is proved, or how best to prove it? This can be checked with Mathematica, but how about a proof?
$$\prod _{n=1}^{\infty } \left(1-\frac{x}{\pi n}\...
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Reference for, and/or proof of, $\prod_{n=1}^\infty(\frac{4n+1}{4n-1})^{4n}(\frac{2n^2-2n+1}{2n^2+2n+1})^n=\sqrt2\cosh(\pi/2)e^{-2G/\pi}$
Context:
I have derived some infinite products that I think are not well known. This is the easiest of them:
$$\prod_{n=1}^{\infty}\left(\frac{4n+1}{4n-1} \right)^{4n}\left(\frac{2n^2-2n+1}{2n^2+2n+1} ...
user1142168
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Is $(-1)^{\infty}=0$? [closed]
I was recently working across the ADT Queue when I thought that such a scenario can be defined by the Grandi series. It is given that:
$$
\sum_{n=0}^k (-1)^n = \frac{1}{2}\big((-1)^k + 1\big)
$$
It is ...
user1070658
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Showing that $\left|\prod_{k=1}^\infty \cos\left(\frac{2\pi}{3^k}\right)\right| > 0$ [duplicate]
I was wondering what would be an easy argument to show that the infinite product $\prod_{k=1}^\infty \cos\left(\frac{2\pi}{3^k}\right)$ does not converge to zero? I understand that the important part ...
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Closed form of a function $f(z)$ whose $\it{only}$ zeros are $n\pi$,$n\pi\omega$ and $n\pi\omega^2$
I need a closed form of a function $f(z)$ whose $\it{only}$ zeros are $n\pi$,$n\pi \omega$ and $n\pi\omega^2 $ where $\omega$ is a cube root of unity and $n\in\mathbb{N}$.
So we need a function of ...
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Evaluating the limit $\lim_{n \to \infty} \left[\log(n) + 2n \log(4n + 2) - \sum_{k = 1}^{2n} (-1)^k (2k+1)\log(2k+1)\right]$
I am trying to show that the limit
$$\lim_{n \to \infty} \left[\log(n) + 2n \log(4n+2) - \sum_{k=1}^{2n} (-1)^k (2k+1) \log(2k + 1)\right] = \frac{2G}{\pi} - \log(4)$$
where $G$ is Catalan's constant.
...
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Limit of $\frac{1}{n!}\prod_{i=1}^{n}(i-\alpha)$
Suppose that $\alpha > 0$. Show that $$\lim_{n\to\infty}\left(1-\frac{\alpha}{1}\right)\left(1-\frac{\alpha}{2}\right)\ldots\left(1-\frac{\alpha}{n}\right)=0$$
Here’s what I’ve tried so far,
$$
\...
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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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What is $\lim\limits_{x\to \infty}[g(x)-g(x-1)]\overset?=$
Hi it's a follow up of $\lim\limits_{x\to \infty}[f(x)-f(x-1)]\overset{?}{=}e$
Let :
$$g\left(x\right)=\int_{0}^{\operatorname{floor}\left(x\right)}\prod_{n=2}^{\operatorname{floor}\left(x\right)}\...
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Infinite product in p-adic fields
I have a (probably simple) question about the convergence of an infinite product in p-adic fields. In particular we consider the completion $\mathbb{C}_p$ of an algebraic closure of the p-adic numbers ...
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is it okay to expand infinite product
For fixed constants $p, q \in \mathbb{R}$, define the function $f: \mathbb{Z}^+ \to \mathbb{R}$ on the even positive integers by
$$ f(x) = \prod_{i = 1}^m \left( p^{2^{k_i}} + q^{2^{k_i}} \right) ,$$
...
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Asymptotics of ratio of q-Pochhammer symbols
In (9.23) of https://arxiv.org/abs/1908.08875 , they claim that it is "easy" to show the following limits $$\lim_{y\to 0}\frac{(yq^\frac{2-r+m}{2};q)_\infty}{(y^{-1}q^\frac{r+m}{2};q)_\infty}...
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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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A question about convergence of functions involving Q-Pochhammer symbols
Consider the function $H:\mathbb{C}\rightarrow\mathbb{C}$:
$$H(s)=\prod_{n=1}^{\infty}\left(1+\frac{1}{2^{ns}}\right)$$
$H(s)$ is convergent on $\Re(s)>0$. Is it absolutely convergent on $\Re(s)>...
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Show $\prod\limits_{k=1}^{+\infty}\coth\left({\frac{\pi k}{2}}\right)=\left({\frac{2}{\pi}}\right)^{\frac{1}{4}}\Gamma\left({\frac{3}{4}}\right)$ [duplicate]
Prove that $$\prod_{k=1}^{+\infty}\coth\left({\frac{\pi k}{2}}\right)=\left({\frac{2}{\pi}}\right)^{\frac{1}{4}}\Gamma\left({\frac{3}{4}}\right).$$
The best I was able to get: $$\ln(P)=\sum_{k=1}^{\...
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Weierstrass definition of the Gamma function
I want to cite Weierstrass's paper on his definition of the Gamma Function. So far I couldn't find any: I went to Wikipedia's page for the Gamma Function and it didn't cite his paper, I went to the ...
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Infinite products $f(x) = \prod_{n=1}^{\infty}(1-x^n)$ and $g(x) = \prod_{n=1}^{\infty}(1+x^n)$
Consider the functions
$
f(x) = \prod_{n=1}^{\infty}(1-x^n)
$
and
$
g(x) = \prod_{n=1}^{\infty}(1+x^n)
$
$f(x)$ is defined for $x\in[-1,1]$ and $g(x)$ is defined for $x\in[-1,0]$.
I was wondering if ...
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If the product of spaces is normal, then factor are normal. [duplicate]
I am proving that if the product space $\prod_{\alpha \in J} X_\alpha$ is normal, then any $X_\alpha$ is normal.
So far I have this:
Let $A$ be closed in $X_\alpha$, then $\pi_\alpha^{-1}(A)$ is a ...
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Question on convergence to zero of an infinite product
Suppose we have two sequences $a_{i}$ and $b_{i}$ such that $a_{i}\in(0,\frac{1}{2})$ and $b_{i}\in (\frac{1}{2},1)$ and $\lim_{n\to \infty} a_{i}=\lim_{n\to \infty} b_{i}=\frac{1}{2}$. I am trying to ...
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Are the coefficients of an infinite product of linear polynomials log concave?
A sequence $\{a_i\}$ is said to be log concave if it satisfies the property that for all $k\in\mathbb{N}$, $$a_{k}a_{k+2}\leq a_{k+1}^2$$
We know that the product of log-concave functions is log-...
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Double infinite product $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$
Question
Compute the products:
$\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$
$\prod_\limits{0<i<j<2020} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\...
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Inverse limit of an inverse system expressed as an infinite product
Let $R$ be a commutative ring, let $S:=R[[x_1, ..., x_n]]$ be the ring of formal power series in multiple variables over $R$ and let $\mathfrak{m}:=(x_1,\dots,x_n)$ be the ideal in $S$, generated by ...
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Stein complex analysis 6.2
Prove that
$$\prod_{n=1}^\infty{n(n+a+b)\over(n+a)(n+b)} = {\Gamma(a+1)\Gamma(b+1)\over\Gamma(a+b+1)}$$
whenever $a$ and $b$ are positive. Using the product formula for $\sin\pi s$, give another proof ...
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Show that $\prod_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$
Show that
$$\prod\limits_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$$
Context:
Inspired by this question, I considered the ...
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What is the product of the areas of every regular polygon inscribed in a circle of area $1$?
What is a closed form of $P=\prod\limits_{k=3}^{\infty}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}\approx 0.05934871...$ ?
This is the product of the areas of every regular polygon inscribed in a ...
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Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that no sequence in $X$ converges to the constant function $1$.
Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that $X$ is a dense subset of $\mathbb{R}^{\mathbb{R}}$ with the product topology and that no sequence in $X$ ...
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Swapping the series and product in the zeta function euler product equality
I was thinking that we have a well known function satisfying the following relation:
$$ \sum_{n\in \Bbb N} h_n(x) =\prod_{\text{p prime}} j_p(x) $$
which is of course the Riemann zeta function, $\zeta(...
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Can $\prod\limits_{k=1}^\infty \left(1- \frac{1}{e^{ \sqrt{2} \pi k}}\right)$ be put into closed form?
Let $$ \alpha = \prod_{k=1}^{\infty} \left(1- \frac{1}{e^{ \sqrt{2} \pi k}}\right) $$
and $$ \beta = \prod_{k=1}^{\infty} \left(1 + \frac{1}{e^{ \sqrt{2} \pi k}}\right) = \frac{\exp \left(\frac{\...
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Is my claim that this number is irrational correct?
Define a number $c$ in the following way: $$c=\ln \left(\prod^\infty_{k=1}\frac{e^{1/k}}{1+\frac{1}{k}}\right)$$ (I can assure you that this converges). Isn't this number transcendental since the ...
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Intuitive explanation for elegant property of the unit circle: product of lengths converges to $2^{-n}$.
In a unit circle, flip the four quarter-circles inside the circle. Draw $n$ line segments from the centre to the flipped quarter-circles, so that the angles between neighboring line segments are equal....
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Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?
On the graph of $y=\tan{x}$, $0<x<\pi/2$, draw $2n$ zigzag line segments that, with the x-axis, form equal-width isosceles triangles whose top vertices lie on the curve. Here is an example with $...
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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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Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.
I'm trying to prove or disprove the following conjecture:
If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$ then $L=\lim\limits_{n\to\infty}\sum\limits_{k=1}^n f\left(\frac{k}{n}\...
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Show that $\lim_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$.
I am trying to show that
$$L=\lim\limits_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum\limits_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$$
Desmos strongly suggests that this is true,...
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Find $C$ such that $\frac{1}{n}\prod_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ converges to a positive number.
I'm looking for the value of $C$ such that $L=\lim\limits_{n\to\infty}\frac{1}{n}\prod\limits_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ equals a positive real ...
- 8,378
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Infinite product of areas in a square, inscribed quarter-circle and line segments.
The diagram shows a square of area $An$ and an enclosed quarter-circle.
Line segments are drawn from the bottom-left vertex to points that are equally spaced along the quarter-circle.
The regions ...
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