Articles of products

Infinite Product is converges

I am adding this problem since it is interesting and valuable to be verified here: Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if $\sum_{k=1}^{\infty} u_k$ converges. What about the inverse problem? Thanks for any ideas.

Gluing together mathematical structures, how?

By structure, I mean that which is defined here: What I’m looking for is a way of gluing together structures so that each structure used is embedded within the whole glued-together object. (Each–meaning not just “most”) I don’t need this embedding to be elementary; just something that “preserves” function, relation, and constant symbols. I […]

Why $\int_0^1(1-x^4)^{2016}dx=\prod_{j=1}^{2016}\left(1-\frac1{4j}\right)$?

In this answer it states that: $\int_0^1(1-x^4)^{2016}dx=\prod_{j=1}^{2016}\left(1-\frac1{4j}\right)$ How to prove that?

Product of two complementary error functions (erfc)

I believe that (i.e., it would be convenient if, and visually appears that) the product of the two complementary error functions: $$\operatorname{erfc}\left[\frac{a-x}{b}\right]\operatorname{erfc}\left[\frac{a+x}{b}\right]$$ will have a solution, or can be approximated with a solution, of a Gaussian form (i.e., $c\operatorname{exp}\left[-\frac{x^2}{2d^2}\right]$, where $c$ and $d$ are functions of $a$ and $b$) when $a>0$ and $c>0$, however I […]

Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?

Suppose that $E_1,E_2$ are two measurable (Lebesgue) subsets of $R^d$. Define $E=E_1\times E_2=\left\{(x,y)|x\in E_1, y\in E_2\right\}$. Can we say that $E$ is a Lebesgue measurable subset of $R^{2d}$? Recall (cf. Stein’s Real analysis) that a subset $E$ of $R^d$ is called measurable if for any $\epsilon>0$, there exist an open set $O\supset E$, such that […]

Infinite product of recursive sequence

Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3…a_\infty$, and possibly find its value?

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with $\gamma$ the Euler-Mascheroni constant, do we have $$\frac{1}{e^\gamma\log x} \prod_{p < x}_{\text{p prime}} \frac{p}{p-1}<1+ \prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{n+1}-1} \ \ ?$$ As $x \to \infty$, the […]

Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.

I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end. Here is what I have so far: $ \prod (1+a_n) &lt \infty \implies \sum a_n &lt \infty $: Similarly we ignore finitely many terms until $|a_n| \leq […]

Proof without induction of the inequalities related to product $\prod\limits_{k=1}^n \frac{2k-1}{2k}$

How do you prove the following without induction: 1)$\prod\limits_{k=1}^n\left(\frac{2k-1}{2k}\right)^{\frac{1}{n}}>\frac{1}{2}$ 2)$\prod\limits_{k=1}^n \frac{2k-1}{2k}<\frac{1}{\sqrt{2n+1}}$ 3)$\prod\limits_{k=1}^n2k-1<n^n$ I think AM-GM-HM inequality is the way, but am unable to proceed. Any ideas. Thanks beforehand.

Properties of Weak Convergence of Probability Measures on Product Spaces

EDIT: For the Bounty, I made a substantial edit revision concerning the structure of the question, to make it more readable (hopefully). Moreover I added a question on problem 2.7 of Billingsley’s book. I have two problems concerning weak convergence of probability measures in product spaces, that arose from Billingsley’s classic “Convergence of Probability Measures” […]