2
votes
1answer
54 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
1
vote
2answers
63 views

Product identities

I need to use the following identities for poisson integral but i can't guz i don't know how to prove them. $$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$ ...
0
votes
2answers
53 views

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate the limit

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$ Clueless on this problem. Seek your help.
1
vote
0answers
50 views

Proving the convergence of a product

I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality: ...
1
vote
1answer
106 views

What does $\displaystyle\prod_{n\geq 1} \frac{n-z}{n+z}$ converge to?

Does the infinite product $$\prod_{n\geq 1} \frac{n-z}{n+z}$$ converge, and if so to what? It seems that $$\lim_{n\rightarrow\infty}\frac{n-z}{n+z} = 1$$ so it is reasonable to think that the product ...
1
vote
0answers
76 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
0
votes
1answer
123 views

Countable product of finite sets with a new metric, compact?

Suppose we have a finite set $E$. Is it true that $E^n$ is compact? The metric on $E^n$ is : $$d(\omega,\omega\prime)=\begin{cases}2^{-\inf \{ n \in \mathbf N:\omega _n \ne \omega'_n\} }&{\omega ...
5
votes
1answer
127 views

Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$

Difficult question from some test somewhere (I forget). $$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$ $x$ is, of course, an integer.
1
vote
1answer
253 views

Homeomorphism of product of topological spaces

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
0
votes
2answers
219 views

the limit of infinite product $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots $

I wonder if the function $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots, 0< y<1$, converges to some well-known function. If we let $ (1+y)(1+y^2)(1+y^3)(1+y^4)\cdots = \prod_{i=1}^\infty (1+y^i) = ...
21
votes
2answers
591 views

Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$

I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
1
vote
1answer
174 views

simplifying a product formula (similar to Euler's sine product)

Can anyone help me out trying to simplify the left hand side of the below equation to obtain the right hand side? $$ \displaystyle\prod_{\substack{n=-\infty \\n\neq ...
2
votes
1answer
220 views

Show that $\prod_{i=1}^n a_i- \prod_{j=1}^n b_i =$ $\sum_{t=1}^{n-1}(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t)$

Pardon the cryptic notation and possibly trivial question. I believe the following holds. Define $$X_t=(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t).$$ Show that ...
3
votes
0answers
166 views

Infinity Product Equality.

Let $\{I_n\}_{n\in\mathbb{N}}$ be a sequence of intervals in the form $$ I_n = \Big [ \frac{q_n}{b_n}, \frac{q_n + 1}{b_n} \Big),$$ where $q_{n}$ is some integer, for all $n\in\mathbb{N}$. Define ...
3
votes
1answer
128 views

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?
7
votes
1answer
204 views

Which is the Abel's theorem invoked in the context of convergence of this infinite product?

Motivation: As I wrote in this answer the following product is evaluated in Proposition 5 of Jean-Paul Allouche and Jeffrey Shallit's paper The ubiquitous Prouhet-Thue-Morse sequence ...
17
votes
4answers
2k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...