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

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2
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0answers
33 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 ...
3
votes
2answers
73 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
38 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
98 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
58 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
37 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
42 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
78 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
40 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
95 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
53 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
35 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
171 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
60 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
19 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
35 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
57 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
107 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
49 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
28 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 ...
9
votes
2answers
314 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
245 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
50 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
82 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
108 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
142 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 $$a_k=\left\{\begin{array}{ll} k & \text{for }k\text{ odd} \\ \frac{1}{k-1} & \text{for }k \text{ even} \end{array}\right..$$ ...
5
votes
0answers
101 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
2
votes
1answer
66 views

Necessary/sufficient conditions for an infinite product to be exactly equal to $1$

Consider an infinite product $$p=\prod_{n=1}^\infty a_n,$$ with $a_n\in\mathbb{R}$ (or $\mathbb{C}$ if possible). Is there an if and only if type theorem for when $p=1$, or is anything known about the ...
1
vote
1answer
56 views

Convergence of product over all primes

How can we find the values of $x$ for which $$\prod_{p \text{ prime}}{1-\frac{x^2}{p^2}}$$ converges? I know that this product $$\prod_{p \text{ prime}}{1+\frac{x^2}{p^2}}$$ converges if and only if ...
1
vote
1answer
61 views

Infinite product whose entries tend to 1 rapidly

Does the following infinite product converge and what is the limit if it exists? $$\prod_{i = 1}^{\infty} \frac{2^i}{2^i+1}$$
3
votes
3answers
85 views

Limits of Topological Vector Spaces

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots ...
0
votes
0answers
19 views

Determining probability of continuous data given predictions

I am doing an investigation into the math used in ScoreAHit's algorithm. In their documentation page they discuss a value they computed dubbed "harmonic simplicity." Their equation for calculating ...
2
votes
0answers
30 views

Countability of unions versus products

Let $D_{n}$ be a set with $2^{n}$ elements for $n=1,2,...$. Let $A = \bigcup_{n=1}^{\infty}D_{n}$, and let $B = \prod_{n=1}^{\infty}\{0,1\}$. Let $A_{k} = \bigcup_{n=1}^{k} D_{n}$, and let $B_{k} = ...
2
votes
0answers
135 views

Equality between an infinite product and an infinite series. How can I reconcile both?

Maybe a trivial question, but how could I reconcile the following equation: $$\displaystyle \prod_{n=2}^\infty \left(\frac{1}{1-\frac{1}{n^2}}\right)^{(-1)^n}=\sum_{n=1}^\infty \left(\frac{1}{(2\,n ...
0
votes
1answer
56 views

absolute convergence of infinite product

We know if $\sum_{n=1}^\infty|z_n|$ converges then $\sum_{n=1}^\infty z_n$ converges absolutely. (kind of trivial) I wonder whether it holds for infinite products, that is, if ...
4
votes
2answers
133 views

Find the product of the series? [closed]

Can anybody please help me to evaluate the following product? $$\prod_{n=2}^\infty\dfrac{n^3 - 1}{n^3 + 1} $$
4
votes
1answer
94 views

Infinite product related to the Wallis product

Some time ago I heard this math question on the radio: The Wallis product $$\frac{2}{1}*\frac{2}{3}*\frac{4}{3}*\frac{4}{5}*\frac{6}{5}*\frac{6}{7}* \dotsb$$ is known to converge to $\pi/2$, but ...
4
votes
2answers
74 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
0
votes
1answer
108 views

Proof of convergence of an infinite product

a) Show that $\Pi_{n=1}^\infty x_n$ converges if and only if for all $\varepsilon>0$ there exists an $N$ such that for all $m\ge n\ge N$, $\left|x_nx_{n+1}\cdots ...
4
votes
2answers
156 views

Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$

What is $\prod\limits_{n\geq 2}(1-\frac{1}{n^3})=?$ $\prod\limits_{n\geq 1}(1+\frac{1}{n^3})=?$ I am sure about their convergence. But don't know about exact values. Know some bounds as well. For ...
2
votes
1answer
47 views

Can someone walk me through this infinite product?

$$\prod_{n=1}^\infty\left(\frac{3}{2^n}\right)^{1/2^n} $$ Sorry if that's not clear, that's $(1/2^n)$ in the superscript. The answer is $3/4$. I've never worked with infinite products before, so a ...
3
votes
2answers
150 views

How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$

How find this $$\prod_{n=2}^{\infty}\left(1-\dfrac{1}{n^6}\right)$$ I think we can find this value have closed form $$\prod_{n=2}^{\infty}\left(1-\dfrac{1}{n^{2k}}\right)$$ since ...
5
votes
2answers
112 views

$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed ...
3
votes
0answers
61 views

Infinite product of sinc functions

Calculate the infinite product $f_q(x):=\prod_{n=0}^\infty\frac{\sin(q^n x)}{q^n x}$, where $x$ is real and $0<q<1$. In other words, $f_q$ must satisfy the functional equation ...
2
votes
3answers
70 views

How do I prove $\lim_{n\to\infty} \frac{n^z n!}{z(z+1)\cdots (z+n)}=\frac{1}{z}\prod_{n=1}^\infty \frac{(1+\frac{1}{n})^z}{1+\frac{z}{n}}$?

How do I prove $$\lim_{n\to\infty} \frac{n^z n!}{z(z+1)\cdots (z+n)}=\frac{1}{z}\prod_{n=1}^\infty \frac{(1+\frac{1}{n})^z}{1+\frac{z}{n}}$$? This could be proven if $$\lim_{n\to\infty} ...
1
vote
1answer
62 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
0
votes
3answers
49 views

Finding limit of $\prod_{t=1}^{n}{\left(1-\frac{2}{(n)(n+1)}\right)^2}$ [duplicate]

Let $$x_n=\left(1-\frac{1}{3}\right)^2\cdot\left(1-\frac{1}{6}\right)^2\cdot\left(1-\frac{1}{10}\right)^2\cdot\left(1-\frac{1}{15}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2 \quad ...
3
votes
1answer
66 views

Comparison between infinite products and series

I need examples of the following facts: 1) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges $\nRightarrow \prod_{j=0}^{+\infty}(1+|a_{j}|)$ converges 2) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges ...