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

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2
votes
1answer
53 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
89 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
votes
1answer
48 views

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

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.
3
votes
3answers
40 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
23 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
votes
0answers
27 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
74 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
52 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$$\ \lim_{x\to \infty} \log \log x \prod_{i< \log x} \frac{p_i -1}{p_i}= \\ \lim_{x\to \infty} \log \log x \prod_{p < \log x}_{p prime} \frac{p-1}{p}$$?
0
votes
0answers
32 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
20 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
28 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
141 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.
1
vote
2answers
82 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
38 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
29 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
votes
0answers
30 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
votes
0answers
69 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 $0<\delta<1$?
1
vote
2answers
123 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
71 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
82 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
78 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
60 views

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

How is it possible to prove the following inequality? ...
6
votes
0answers
154 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
79 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
45 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
105 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
63 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
38 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 ...
1
vote
2answers
44 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
83 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
45 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
98 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
vote
3answers
67 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
37 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
173 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
65 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
37 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
20 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
41 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 ...
4
votes
0answers
64 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
114 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
52 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 ...
2
votes
1answer
29 views

Normal convergence of $\prod_{n\geq n_{0}}1/f_{n}$

I am having a hard time trying to study complex analysis for one of my exams. I have some problems with an exercise that I am supposed to solve. The setting is the following. Let $D$ be a domain ...
14
votes
4answers
606 views

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
2
votes
2answers
270 views

Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $

I am revising Complex Analysis and I am a bit confused. I have a couple of results from lectures which say that $\prod_{n=1}^\infty (1+a_n)$ converges if and only if the sum $\sum_{n=1}^\infty ...
1
vote
2answers
55 views

Rate of growth of an Euler Product

Merten's Theorem gives $$ \prod_{p < x} \left( 1 - \frac{1}{p} \right) \sim k(\log x)^{-1} $$ I also know that $$ \prod_{p < x} \left(1 - \frac{n}{p} \right) \leq \prod_{p < x} \left(1 - ...
3
votes
1answer
89 views

Can the exact value of the product over the Riemann zeta function at even arguments be evaluated?

According to wolframalpha, the product over the Riemann zeta function at even arguments converges : $$\prod_{n=1}^{\infty} \zeta(2n) \approx 1.82 $$ Q1: Can it be proved that this product actually ...
3
votes
4answers
114 views

Why does $\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverge?

$\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverges, and I have no idea why? It would seem using L'hop, $\frac{n^3 + n^2 + n}{n^3 + 1}$ goes to 1. So it should end up just being ...
4
votes
2answers
153 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
2
votes
1answer
56 views

Why does this product diverge?

Consider the partial product $$ p_n=\prod_{k=1}^{2n}a_k,$$ where $$a_k=\left\{\begin{array}{ll} k & \text{for }k\text{ odd} \\ \frac{1}{k-1} & \text{for }k \text{ even} \end{array}\right..$$ ...