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

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1answer
26 views

How to establish the convergence?

How to show that $\prod\limits_{n=1}^\infty \frac{1+x^{\delta^n}}{2}$ is convergent if $x>0, 0<\delta<1$ ? Thanks in advance
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2answers
26 views

How to establish the convergence of this infinite product?

How to prove that $\prod\limits_{n=1}^\infty \cos^2(\frac{1}{n^2})$ is convergent ? I have got the discussion here but could not figure out if we are asked to prove the convergence, through simple ...
2
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2answers
46 views

Proving $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula

I am trying to show that $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula: $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\limits_{n=1}^\infty\left(1+\frac{z}{n}\right)^{-1}e^{z/n}$$ where $\gamma$ ...
1
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0answers
21 views

Special values of the Dedekind eta function

Does anyone know of an article or a book which contains a comprehensive list of some known explicit values of the Dedekind eta function $\eta(\tau)$?
3
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1answer
67 views

Convergence of infinite product of prime reciprocals?

Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
3
votes
1answer
65 views

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

I'm interested in finding a closed form for $$\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$$ Wolfram Alpha confirms that it converges, but I can't find any plausible closed forms. I've made ...
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0answers
52 views

Infinite product: Problem I

What is the closed form value of the infinite product $$ ...
18
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5answers
446 views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
2
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1answer
65 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
2
votes
2answers
107 views

$\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Let $A_n$ be independent events with $P(A_n) \neq 1$. Show that $\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$ It kind of looks obvious but I really have no idea how to prove it. Can ...
0
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1answer
63 views

Calculate $\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$ [on hold]

Calculate $$\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$$ Any suggestions please? Thank you very much.
4
votes
3answers
46 views

$\prod\left(1-p_n\right)>0$

I want to prove that if $0\le p_n<1$ and $\sum p_n<\infty$, then $\prod\left(1-p_n\right)>0$ . There is a hint : first consider the case $\sum p_n<1$, and then show that ...
0
votes
3answers
24 views

Quotient of infinite products

I have a very simple question, but I never learned about infinite products and now have to use them. Am I right in assuming that $$ {\prod_{k=1}^{\infty} f(k)\over\prod_{k=1}^{\infty} ...
0
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0answers
32 views

Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and ...
5
votes
2answers
84 views

How can we apply the Borel-Cantelli lemma here?

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically ...
0
votes
1answer
55 views

As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$? [duplicate]

How is$$\ \lim_{x\to \infty} \log \log x \prod_{i< \log x} \frac{p_i -1}{p_i}= \\ \lim_{x\to \infty} \log \log x \prod_{p < \log x}_{p prime} \frac{p-1}{p}$$?
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0answers
33 views

Function $f(z)=\frac{\sin \pi(z-\lambda_n)}{\pi(z-\lambda_n)}$ and infinite product.

What is the relationship between the infinite product $$\prod_n \left(1-\left(\frac{z}{\lambda_n}\right)^2\right), \ \ \ \ \ z\in \mathbb C, \lambda_n\in \mathbb R$$ and the function $$f(z)=\frac{\sin ...
1
vote
1answer
21 views

Good lower bound on an infinite product

I am trying to find a good lower bound on $\prod_{k = j}^\infty (1- C \cdot 2^{-k})$, where $C$ is constant, in terms of $j$ that goes to $1$ as $j\rightarrow \infty$. Does anyone know of any ...
0
votes
1answer
31 views

If an infinite product$\ P= 0$ times a constant and an unbounded function gives$\ 1$, will the same for$\ Q< \infty$ certainly be$\ >1$?

Let$\ P=\lim_{n\to \infty} \prod_{i=1}^n x_i=0$, $\ Q=\lim_{n\to \infty} \prod_{i=1}^n y_i< \infty$ and$\ \lim_{n\to \infty} f(n)= \infty$. If, for some real$\ k$, $\ \lim_{n \to \infty} k f(n) ...
3
votes
3answers
146 views

Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

I'm less than a novice in analysis, I don't even know how to approach this. Thanks in advance.
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2answers
83 views

Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$?

I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on Wolfram|Alpha. Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance.
1
vote
1answer
41 views

Limit of a product

I need to find the value of $$L=\lim_{n \rightarrow \infty}\displaystyle\prod_{r=1}^{n} \left(1+\dfrac{r^2}{n^2}\right)^{1/n}$$ Is doing this OK?-- $$\begin{align} L &=\lim_{n \rightarrow ...
1
vote
0answers
37 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
2
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0answers
35 views

Interchange of infinite product and limit

The Problem Let $(a_{n,m})_{n,m\in \mathbb{N}}$ be an sequence of complex numbers. Under which conditions can I interchange product and limit? $\lim_{m\to\infty}\prod_{n=1}^{\infty} ...
4
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0answers
71 views

Limit of infinite product

Is it possible to find an analytic form for the limit of the infinite product: $$ \prod_{n=1}^\infty\frac{1+x^{\delta^n}}{2} $$ where $ x>0 $ and $0<\delta<1$?
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vote
2answers
128 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
2
votes
3answers
74 views

Landau's proof that $\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}$

In the Handbuch, Landau proves that for all $s>1$ the following equality holds $$\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}.$$ I'm having trouble with the following part of his proof: Landau says that ...
0
votes
4answers
83 views

Can this limit be solved in this manner?

$\lim _{n\to \infty }\left(\left(1+\frac{1}{1}\right)\cdot \left(1+\frac{1}{2}\right)^2\cdot ...\cdot \left(1+\frac{1}{n}\right)^n\right)^{\frac{1}{n}}$ Hi. I'm trying to solve this limit without ...
0
votes
2answers
80 views

Is it true that $\prod_{i=1}^{\infty} a_{i,t} \rightarrow \prod_{i=1}^{\infty}L_i $ when $a_{i,t} \rightarrow L_i$ for every $i$, and $L_i\to 1$?

Let $A_1,A_2,A_3,\dots$ be a sequence of sequences where each $$A_i = a_{i,1},a_{i,2},a_{i,3},\dots$$ Each sequence $A_i$ converges and in particular as $t \rightarrow \infty$, $a_{i,t} \rightarrow ...
1
vote
1answer
61 views

Proof of an inequality involving $(N-1)!$

How is it possible to prove the following inequality? ...
6
votes
0answers
159 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
2
votes
2answers
80 views

Does this product converge?

Does this product converge? $$\prod_{x=0}^{\infty}\frac{(30x+5)(30x+9)(30x+15)(30x+21)(30x+27)}{(30x+7)(30x+13)(30x+19)(30x+23)(30x+31)}$$ I have tried solving this by hand, and it seems to get ...
4
votes
2answers
48 views

Closed form expression for an infnite product

I am interested in a closed form expression for the limit of the sequence $(a_n)$ where \begin{equation} a_n = \prod_{k=1}^n (1 - \tfrac{c}{k}) \end{equation} where $c$ is not equal to $1$ and is ...
1
vote
1answer
106 views

Playing fast and loose with divergent series [closed]

I have been playing around recently with the regularization of infinite divergent sums and products, e.g. $$1+1+1+1+1+\ldots=\zeta(0)=-\frac{1}{2}$$ $$1+2+3+4+5+\ldots=\zeta(-1)=-\frac{1}{12}$$ ...
3
votes
1answer
66 views

An ultrafilter product topology

Suppose $X=\prod _{i\in\omega}X_i$ is the cartesian product of topological spaces $X_i$ and $u$ is a filter on $\omega$. Define a basis for $X$ by taking the collection of all sets of the form ...
3
votes
1answer
41 views

How to calculate this expression?(multiplication)

How do I show that for any starting $n$? (I am not really sure it is $0$, but I think it is). $$\prod_{i=n}^\infty \left[1-\frac 1 i\right]=\prod_{i=n}^\infty \left[\frac{i-1}{i}\right]=0$$ I tried ...
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vote
2answers
45 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
4
votes
1answer
84 views

Infinite product $1+1/k^3$ [duplicate]

Ramanujan's notebooks contain the result $$\prod_{k=1}^{\infty} \Big( 1 + \frac{1}{k^3}\Big) = \frac{1}{\pi} \mathrm{cosh}\Big( \frac{\pi \sqrt{3}}{2}\Big).$$ It doesn't seem like this is proved there ...
0
votes
1answer
47 views

An Inequality Involving Prime Numbers

Let $p_i$ be the $i^{th}$ prime number. It seems as though the following inequality is true for all positive integers $m$ and real numbers $x>1$: ...
4
votes
3answers
100 views

Is there an easy way of proving $\prod_{k=1}^\infty \cos(x/2^k) = \sin(x)/x$?

I just answered this question distribution of infinite sum of $\sum (2x_n -1)/2^n$ by using the formula in the title which I lifted off a random formula sheet on the internet. My question is, how ...
1
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3answers
69 views

Infinite series convergence test

Test the convergence of the following series: $${\sqrt{n+1}-1\over (n+2)^3 -1} +... \infty$$ (This is a problem I got on my test today, I constructed a similar series without the -1 part and showed ...
1
vote
1answer
39 views

Is there an analog of the p-series test for infinite products?

What I mean: P-series: $$\sum_{n=1}^\infty\frac{1}{n^p}$$An infinite product "P-series": $$\prod_{n=1}^\infty(1+\frac{1}{n^p})$$ For what $p\in\mathbb{R}$ does the infinite series converge? Diverge? ...
4
votes
2answers
178 views

Find the limit of $\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$

Find the limit of $$\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$$ The limit does exist, but I can not get it. Thanks Willie-Wong & Lee Mosher for correcting the expression.
0
votes
1answer
67 views

Infinite product convergence for cosine

I have trouble proving the following: if $$\sum_k^\infty|z_k|^2 < \infty$$, then $$\prod_k^\infty \cos(z_k)$$ converges. (Note $z_k$ are complex numbers). I think some relevant proof of convergence ...
0
votes
0answers
38 views

Evaluating $\lim_{n\rightarrow \infty }\prod_{i=1}^{n}\frac{1}{ab^{i-1}+1}$

I'm sure that this function must converge to a constant but I can't write it in a closed form. $$\lim_{n\rightarrow \infty }\prod_{i=1}^{n}\frac{1}{ab^{i-1}+1}$$ $a>0$, $0<b<1$, ...
0
votes
0answers
23 views

Infinite Cartesian product - notation for inverse image

Suppose we have an infinite Cartesian product: $$\displaystyle \prod_{i\in I} X_i =\{f:I\rightarrow \bigcup_{i\in I}X_i : \forall i \in I (f(i)\in X_i)\}. $$ Denote by $\pi_j$ the natural projection ...
0
votes
1answer
43 views

trouble with infinite values from exp() and log()

I'm writing a function for Gaussian mixture models with spherical covariance structures--ie $\Sigma_k = \sigma_k^2 I$. This particular function is similar to the ...
5
votes
0answers
66 views

About $\prod{\left(1-q^n\right)^{5}}$

Is there a result about the non-vanishing of coefficients of $$\prod_{n=1}^{+\infty}{\left(1-q^n\right)^{5}}=1-5q+5q^2+10q^3-15q^4-6q^5-5q^6+25q^7+15q^8-20q^9+\cdots \text{ ?}$$ Thanks !
3
votes
2answers
119 views

A product for 1/e?

This question is related to this question but I see that one part is really not a statistics question. That $\lim_{n \to \infty} (1 - 1/n)^n = 1/e $ is clear. What is not clear to me is under what ...
1
vote
1answer
54 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...