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

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38 views

Showing that $\prod_{k=1}^\infty \cos\left(\frac{2\pi t}{3^k}\right)$ does not vanish at infinity.

As the title says, I need to show that $f(t)=\prod_{k=1}^\infty \cos\left(\frac{2\pi t}{3^k}\right)$ does not vanish at infinity, i.e. $f\not \in C_0(\mathbb{R})$. I tried looking at the series $$ \...
4
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3answers
135 views

Closed form for $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$

Is there any closed form for the infinite product $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$? I think it is convergent for any $x\in\mathbb{R}$. I think there might be one because there is a closed form ...
2
votes
0answers
68 views

Question about an infinite product

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
2
votes
0answers
46 views

An infinite product for $\arccos \alpha$ similar to Viete's formula for $\pi$

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
1
vote
2answers
39 views

Infinite product of sinc function

Is there a closed form for this infinite product? : $$\prod_{n=1}^{\infty}\operatorname{sinc}\left(\frac{\alpha}{n}\right)$$ where $\operatorname{sinc}(x)=\sin(x)/x$ is the familiar "sinc" function ...
11
votes
1answer
178 views

Closed form of infinite product $\prod\limits_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$

I encountered this infinite product while solving another problem: $$P(x)=\prod_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$$ $$P(x)=P \left( \frac{1}{x} \right)$$ I strongly ...
0
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0answers
15 views

Upper bounds for the modulus of $f(s)=\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}s\right)$

Let the complex variable $s=x+iy$, and $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisors function (is a known multiplicative function in number theory, for example $\sigma(1)=1$ and $\sigma(6)=1+2+3+6=...
4
votes
2answers
120 views

Showing $\int_{0}^{1}{x^{2n}-x\over 1+x}\cdot{dx\over \ln{x}}=\ln\left({2\over \pi}\cdot{(2n)!!\over (2n-1)!!}\right)$

Integrate $$I=\int_{0}^{1}{x^{2n}-x\over 1+x}\cdot{dx\over \ln{x}}=\ln\left({2\over \pi}\cdot{(2n)!!\over (2n-1)!!}\right)\tag1$$ $${x^{2n}-x\over 1+x}=\sum_{k=0}^{\infty}(-1)^k(x^{2n}-x)x^k\...
5
votes
2answers
97 views

Prove ${\Gamma^2(1/4)\over 4\sqrt{2\pi}}=\prod_{n=1}^{\infty}\left({2n+1\over 2n}\right)^{(-1)^{n+1}}$

Integrate $$I=\int_{0}^{\infty}{1-e^{-x}\over 1+e^{2x}}\cdot{dx\over x}={\ln{\Gamma^2(1/4)}\over 4\sqrt{2\pi}}\tag1$$ By Frullani's theorem $$I=\sum_{n=1}^{\infty}(-1)^{n+1}\int_{0}^{\infty}{e^{-...
0
votes
2answers
36 views

Algebraic simplifications

Given the following line of maths: $$(1-x)(1+x)(1+x^2)(1+x^3)...(1+x^{2^n})$$ we can simplify it: $$(1-x^2)(1+x^2)...(1+x^{2^n})$$ and so on to obtain: $$(1-x)(1+x)(1+x^2)(1+x^3)...(1+x^{2^n}) = (...
3
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1answer
54 views

What are practical applications of infinite products?

My analysis book covers a section on infinite products. So I started wondering what the practical applications of infinite products are in science and engineering, but couldn't find anything yet. Also,...
0
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1answer
24 views

Principal argument summation

Let $\text{Arg}$ be a an principal argument in $(-\pi, \pi]$. I know that, for all $z_1,z_2\in\mathbb{C}\setminus \{0\}$, the expression $\text{Arg}(z_1z_2)= \text{Arg} z_1 + \text{Arg} z_2$ doesn't ...
2
votes
1answer
69 views

Computing an infinite product $\prod_{n=1}^{\infty} \frac{1}{2}(1+\cos\frac{x}{2^n})$

I would like to compute the infinite product $\displaystyle f(x)=\prod_{n=1}^{N\rightarrow\infty} \frac{1}{2}\left({1+\cos\frac{x}{2^n}}\right)$ for a given real $x$. Since the terms in the product ...
0
votes
1answer
26 views

Uniformly convergence on a set implies a smaller set

Here is a product, that looks like this $\prod_{n=1}^{\infty}p_n(z)$. There are two questions: If you have shown that the product is uniformly convergent on a compact subset of $\mathbb{C}$, is ...
2
votes
1answer
55 views

Does $\frac1n\sum\limits_{k=1}^na_k^2\to\rho$ with $0\le\rho<1$ imply$\prod\limits_{k=1}^na_k\to0$?

Let $\{a_n\}$ be a sequence of real numbers such that $\lim_{n\to \infty}\frac{\sum_{j=1}^n a_j^2}{n}=\rho$, and $0\le\rho<1$. The goal is to check whether the following is true $$\lim_{n\to \infty}...
0
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1answer
25 views

Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
2
votes
1answer
66 views

Prove that $\prod_{k=1}^{\infty}\left(1-e^{-\pi{k\sqrt n}}\right)=\frac{\sqrt2 g_n^2\psi(e^{-\pi\sqrt n})}{e^{\frac{\pi\sqrt n}{12}}}$

The source of idea is from here Where $$\psi(q)=\sum_{k=0}^{\infty}q^{\frac{k(k+1)}{2}}$$ Prove that, (1) $$\prod_{k=1}^{\infty}\left(1-e^{-\pi{k\sqrt n}}\right)=\frac{\sqrt2 g_n^2\psi(e^{-\pi\...
1
vote
0answers
11 views

How to express a homogenous function using an infinitely recursive matrix operation?

As is well known, if the multivariate function $f(\mathbf{x})$ is homogenous of degree $h$, then the partial derivatives of $f$ are homogenous of degree $h-1$. Also, say that we know $f$ is ...
0
votes
1answer
28 views

Does this infinite product converge? And how to express it neatly?

First of all, does this product have a "nicer" functional form--i.e., analogous to how you can write geometric sums in a nice closed expression: $$(x-0)(x-1)(x-2)...(x-n)$$ Secondly, does this ...
7
votes
2answers
92 views

Showing that $\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{2^n+1}L_{2^n+1}}\right)=\frac{3}{\phi^2}$

Infinite product $F_{n}:=[1,1,2,3,5,8,\cdots]$ and $L_{n}:=[1,3,4,7,\cdots]$ for $n=1,2,3,\cdots$ respectively. $\frac{1+\sqrt5}{2}=\phi$ Show that, $$\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{...
3
votes
2answers
75 views

How do I evaluate $\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.
1
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0answers
88 views

Compute the order, type and genus of the entire function $\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}z \right) $

Since $$\sum_{n=1}^\infty\frac{1}{(n^3/\sigma(n))}=\frac{\pi^2}{6}\zeta(3)$$ converges, where $\sigma(n)$ is the sum of divisor function (with maximal size a constant times $n\log\log n$), and $\zeta(...
13
votes
1answer
178 views

Prove $\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$

Infinite product We proposed (1) $$\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$$ How one go about proving (1) ? take ...
3
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1answer
67 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha )|\prod_{n=0}^{\infty}\left[1+\...
2
votes
1answer
40 views

Can you provide us a good approximation for $\sum_{n=1}^{\infty} \left| \log \left( 1+\frac{\mu(n)}{n^2} \right) \right|$?

Let $a_n=\frac{\mu(n)}{n^2}$, where $\mu(n)$ is the Möbius function. Since $\sum \left| a_n \right| $ is convergent by the comparison test, then a proposition from analysis ensures that $$\mathcal{S}=\...
2
votes
1answer
46 views

Finding the value of an infinite product

Find the value of the product : $$P=\sqrt{\frac12}\sqrt{\frac12+\frac12\sqrt{\frac12}}\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}\ldots$$ This was asked in an exam yesterday, I ...
2
votes
2answers
35 views

$q$-digamma function evaluation

What is the value of $\psi_2^{(0)}(1)$, where $\psi_q^{(0)}(z)$ is the $q$-digamma function? My attempt: \begin{align*} \psi_2^{(0)}(z) &=\frac{1}{\Gamma_2(z)}\frac{d\Gamma_2(z)}{dz} \\&=\...
1
vote
1answer
25 views

If and only if condition for a product space to be hausdorff.

Is this true $\forall i \in I \ X_i $ is hausdorff $\iff \prod_{i \in I}X_i $ is hausdorff. I understand that $(\rightarrow )$ is true but don't know if $(\leftarrow)$ is true. If the other ...
2
votes
1answer
41 views

Expanding an infinite product of infinite series

Here's a fragment of something I posted in an answer a few months back: \begin{align} & \left( 1 + \frac 1 {a_1} + \frac 1 {a_1^2} + \frac 1 {a_1^3} + \cdots \right) \\ \times {} & \left( ...
4
votes
3answers
77 views

Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$

The question Prove that: $$\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$$ What I've tried Knowing that: $$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)...
2
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0answers
54 views

What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ...
4
votes
2answers
57 views

Showing that $\sigma=\prod_{n=1}^{\infty}(n!)^{\frac{1}{2^{n+1}}}$

Somos's quadratic recurrence constant The Somos's Quadratic recurrence constant is defined by the sequence $g_n=ng_{n-1}$ with initial value of $ g_0= 1$ The value of $\sigma=1.661687...$ An ...
0
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1answer
58 views

$\cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$?

I need to show that $$ \cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$$. For $x \notin \mathbb Z$ one can use $\cos(\pi x ) = \frac{\sin(2 \pi x )}{2 \sin(\pi x )}$ ...
3
votes
0answers
43 views

Can this Gamma function expression be reduce to its lowest form?

Question 1, Can this Gamma expression be further simplified to its lowest form? $$\frac{a^2\Gamma\left(1+\frac{1}{a}\right)\Gamma\left(a-\frac{1}{a}\right)}{\Gamma^2\left(\frac{1}{a}\right)\Gamma\...
2
votes
1answer
32 views

Show that, $\prod_{k=2}^{\infty}\left(\prod_{n=0}^{\infty}(n!)^{\frac{k-1}{k^{n+1}}}\right)=\prod_{m=2}^{\infty}m^{\zeta(s)-1}$

The factorial n! is defined for all positive integer n as $$n!=n(n-1)(n-2)\cdots2\cdot1$$ The $\zeta(s)$ is defined for all real R(s)>1 as $$\zeta(s)=\sum_{n=0}^{\infty}\frac{1}{(n+1)^s}$$ Show that,...
2
votes
2answers
49 views

Closed form for $\prod_{i=0}^{\infty}(1+x^{2^i})$

I've recently come across the infinite product $\prod_{i=0}^{\infty}(1+x^{2^i})$ and I was wondering if there is a closed form expression for this, or even if it diverges for all non-zero $x$. Thanks!...
0
votes
1answer
65 views

Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. [duplicate]

Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. Remark: I know that $\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$, but not how to prove that, I totally stalled.
2
votes
0answers
28 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
3
votes
1answer
90 views

Infinite product include summation

I would like to find an infinite product of$$\prod _{n=2}^{\infty} \left(1+\frac{(-1)^{n-1}}{a_n}\right)$$ where $a_n = \sum_{k=1}^{n-1} \frac{n!(-1)^{k-1}}{k!} $ I tried to compute $a_2 , a_3 ,...$,...
-2
votes
1answer
50 views

Infinite Sets Proof [closed]

I have a few questions regarding this problem below: Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|. Would I assume that |A| = |B|? Which would obviously make A ≈ B ...
1
vote
0answers
27 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
3
votes
1answer
60 views

Prove that $ζ(4)=π^4/90$ knowing that $\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$

The question Knowing that: $$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right) \tag{1}$$ obtain the Taylor series expansion of $\frac{\sin(πz)}{πz}$ to deduce: $$ \sum_{1 ≤ n_1 < n_2 &...
0
votes
0answers
49 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
2
votes
1answer
60 views

Knowing the representation of $\frac{π^2}{\sin^2(πz)}$, deduce the Taylor series of $\frac{z^2}{\sin^2{z}}$

The question Consider the representation: $$ \frac{π^2}{\sin^2(πz)} = \sum_{n∈ℤ}\frac{1}{(z+n)^2} \tag{0}$$ valid for all $z ∈ ℂ \setminus ℤ$. Deduce that the Taylor series of $z^2/\sin^2(z)$ for $|z|...
2
votes
0answers
78 views

Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\Omega_k, \mathcal{F}_k)$. Consider the product measures on $(\prod_{k=1}^...
1
vote
1answer
40 views

Show that $\frac{z}{1-z} = \sum_{j=0}^∞ \frac{2^j}{1 + z^{-2^j}}$ when $z ∈ \mathbb{D}$

The question Knowing that with $z ∈ \mathbb{D}$: $$ \prod_{k=0}^∞(1 + z^{2^k}) = \frac{1}{1-z} $$ prove that with $z ∈ \mathbb{D}$: $$ \sum_{j = 0}^∞ \frac{2^j}{1 + z^{-2^j}} = \frac{z}{1-z} $$ ...
6
votes
3answers
391 views

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\...
1
vote
0answers
23 views

Understanding the expansion of product notation.

I have a question regarding the expansion of product notation in the picture below. Equation 3.1 in the attached picture is $$\prod_{n=1}^{\infty}(1+zq^n)(1+z^{-1}q^{n-1})=\frac{1}{\prod_{n=1}^{\...
4
votes
3answers
66 views

Evaluate this infinite product: $\prod_{n=3}^{\infty} \left(\;1-\frac{4}{n^2}\;\right)$

$$\prod_{n=3}^{\infty} \left(\;1-\frac{4}{n^2}\;\right)\;=\;\text{???}$$ I took the LCM and split the numerator as $(n+2)(n-2)$ and then took the product of the numerator and the denominator ...
-2
votes
3answers
51 views

Is there another notation for this set-theoretic formula? [closed]

I am writing a book. In my drafts there are formulas like $\prod_{X\in S} X$ or $\{ \operatorname{im} P \mid P\in\prod_{X\in S} X \}$. If there are other way to write the same expressions, I should ...