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

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2
votes
1answer
59 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 ...
-4
votes
0answers
36 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
1
vote
0answers
8 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
23 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 ...
4
votes
2answers
56 views

Show 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, ...
0
votes
1answer
48 views

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

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 $.
0
votes
0answers
22 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 ...
1
vote
0answers
38 views

Proving the uniform convergence of series from the infinite product

I have been trying to prove that this infinite product $(1)$ is uniformly convergent on every compact subsets of $\mathbb{C}\setminus\mathbb{Z}^-$. $$\prod_{m=1}^\infty \frac{\left ( \frac{1}{m}+1 ...
13
votes
1answer
162 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
votes
1answer
65 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 ...
2
votes
1answer
36 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 ...
2
votes
1answer
42 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
31 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
22 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 ...
0
votes
1answer
34 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
72 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} ...
2
votes
0answers
51 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
55 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
votes
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
41 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? ...
1
vote
1answer
29 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 ...
2
votes
2answers
48 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 $x$. ...
0
votes
1answer
63 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
82 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
47 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
26 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
58 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
47 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 ...
2
votes
0answers
75 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 ...
1
vote
1answer
38 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
378 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: ...
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 ...
3
votes
3answers
62 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
48 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 ...
1
vote
1answer
38 views

Proof of Jacobi triple product by taking the limit

Assume that we know $$ \prod_{k=1}^{n}(1+q^{2k-1}z)(1+q^{2k-1}z^{-1})=C_{0}+\sum_{k=1}^{n}C_{k}(z^k+z^{-k}), $$ with $$ ...
4
votes
1answer
84 views

Euler Product formula for Riemann zeta function proof

In class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ ...
10
votes
3answers
331 views

Product of all real numbers in a given interval $[n,m]$

READ-ME I have now what I can call for myself answers to all my problems and subquestions proposed in this post, thus I accepted Strings answer as the answer to this question since it was of most ...
1
vote
2answers
72 views

Proving a result in infinite products.

We assume that $\sum |a_n|^{2}$ converges, then I want to conclude that $\prod (1+a_n)$ converges to a non zero element $\iff$ the series $\sum a_n$ converges. My attempt If $\prod (1+a_n)$ ...
2
votes
1answer
110 views

Prove that $\lim_{\ r\ \to \ \infty} \dfrac{r! r^x}{x(x+1)(x+2) \dots (x+r)} = \int_{0}^{\infty} t^{x-1} e^{-t} dt $

From Havil & Dyson, "Gamma: Exploring Euler's Constant", section 6.1 I can't prove the following Euler's theorem : ... on 13 October 1729, Euler had already proposed to Goldbach the ...
1
vote
4answers
203 views

If $a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$ then $\lim_{n\to\infty}a_n=?$

I have an objective type question:- If $$a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$$ then $\lim_{n\to\infty}a_n=?$:- A)$0$ B)limit does not exist ...
1
vote
1answer
49 views

Show $\cos \pi z=\prod\limits_{-\infty}^\infty \left(1-\frac{2z}{2n-1}\right) e^{\frac{2z}{2n-1}}$

I have already shown $\cos \pi z=\prod\limits_{n=1}^{\infty}\left(1-\frac{4z^2}{(2n-1)^2}\right)$. Here is what I have after that, \begin{equation*}\begin{split} \cos(\pi z) & = ...
31
votes
3answers
800 views

Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln ...
2
votes
3answers
185 views

What is $2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty$ equal to?

I came across this question while doing my homework: $$\Large 2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty=?$$ $$\small\text{OR}$$ ...
0
votes
0answers
40 views

Infinite expansion of difference of squares?

I was just wondering if the following is legitimate or if there's a problem with it: Using difference of squares, $a^n - b^n = (a^{n/2}-b^{n/2})(a^{n/2}+b^{n/2}) = ...
0
votes
1answer
11 views

How to prove the product representation of Baenes G-function

How to prove this formula of Barnes G-dunction $$G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z(z+1)}{2}- \frac{\gamma z^{2}}{2}\right)\, ...
1
vote
3answers
35 views

Rearranging infinite product

I know that $$\frac{\sin x}x=\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right).$$ Why exactly can I take the product and factor $x^2$? $$\prod_{n=1}^\infty ...
1
vote
0answers
25 views

Zeroes of infinite product function?

Given an entire function $\prod_{n=0}^∞ E_n (\frac{z}{z_n})$ , show that $(z_n )_n∈N$ is a complete list of the zeroes of this function in which each zero appears as many times as its multiplicity. ...
6
votes
1answer
288 views

What is the infinite product of (primes^2+1)/(primes^2-1)?

I have shown that the infinite product $$\prod_{p \in \mathcal{P}}\frac{p^2+1}{p^2-1}$$ is equal to $\frac{5}{2}$ (pretty remarkable!). I have checked this numerically with Wolfram Alpha for up to ...