Articles of infinite product

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.

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 nature of the $a_n$ when $p=1$ ? Clearly one case springs to mind: $a_n=1$ for all $n\geq 1$. Is this the only case or are […]

Weierstrass factorization theorem and primality function

I’m interested in application of the Weierstrass factorization theorem to the primality function. Let $np(x)\colon \mathbb N\to \mathbb N$ is a “not-prime” function: $$ np(x) = \begin{cases}1, & \text{$x$ is not prime}\\0, & \text{$x$ is prime}\end{cases} $$ Obviously, prime numbers are zeros of $np$. Therefore it seems reasonable to use Weierstrass factorization theorem to express […]

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

Assume $k$ positive integer. How can I show that $$ \tag 1 \prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}? $$ I know that $$ \tag 2 \underset{n\geq1}{\prod}\left(1-\frac{k^{2}}{n^{2}}\right)=\frac{\sin\left(\pi k\right)}{\pi k}$$ and so if $k$ is an integer the product is $0$ but how can I use these information?

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?

Question on $\Pi_{n=1}^\infty\left(1-\frac{x^a}{\pi^an^a}\right)$ and the Riemann Zeta function

Question on what is $$\Pi_{n=1}^\infty\left(1-\frac{x^a}{\pi^an^a}\right)$$ I deduced that for any $a=2,4,6,\dots$, the above product is simplify-able into a product of sines, $$\Pi_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right)=\frac{\sin(x)}x$$ $$\Pi_{n=1}^\infty\left(1-\frac{x^4}{\pi^2n^2}\right)=\frac{\sin(x)\sin(xi)}{x^2}$$ And trying to put this into a general form, I got $$f(x):=\frac{\sin(x)}{x}$$ $$\implies f(x)=f(-x)$$ $$f(x)f(ix)=\Pi_{n=1}^\infty\left(1-\frac{x^4}{\pi^4n^4}\right)=\sqrt{f(xe^{\frac{\pi i}{2}})f(xe^{\pi i})f(xe^{\frac{3\pi i}{2}})f(xe^{2\pi i})}$$ Using $(a+b)(a-b)=a^2-b^2$, we can get something along the lines of $$\Pi_{n=1}^\infty\left(1-\frac{x^{2^a}}{\pi^{2^a}n^{2^a}}\right)=\sqrt{f(xe^{\frac{1\pi i}{2^a}})f(xe^{\frac{2\pi i}{2^a}})\dots […]

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 \log(1+a_n) $ converges absolutely. And also, the infinite product converges absolutely if and only if the $|a_n|$ are summable. Consider the example: $$\prod_{n=2}^\infty […]

Divergent products.

My question are about divergent products. I’m a Dutch student so i may lack the skil to write it down in the correct notation and forgive my spelling errors. A thing i’ve found on the internet was that $$\frac{\sqrt{2 \pi}}{\Gamma(n)} = \prod_{c=0}^{\infty} (c+n) $$ according to my own numerical foundings should be true but i’ve […]

A tricky infinite sum— solution found numerically, need proof

Consider an infinite sum of the following form: $X Y^{\alpha} + X^2 Y^{\alpha + \alpha^2} +X^3 Y^{\alpha + \alpha^2 + \alpha^3} + …$ …which can be expressed more succinctly as: $\sum\limits_{j = 1}^{\infty}X^j \prod\limits_{k = 1}^{j}Y^{\alpha^k}$ …where $0 < X < 1$, $0 < \alpha < 1$, and $Y > 0$. Using numerical methods, I […]

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 asking to apply the Borel-Cantelli. Yet the suggested solution went differently: via $\prod_{n=1}^\infty \mathbb P( A_n^c)=0$. How can we apply the Borel-Cantelli lemma here? I.e. how to show […]