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

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4answers
89 views

Infinite sum of products on my mid-term

This problem was on my calculus mid-term : Determine the convergence or divergence of the series $$ \sum\limits_{n = 1}^\infty {\prod\limits_{k = 1}^n {\frac{{4k - 3}}{{4k - 1}}} } $$ I tried ...
2
votes
1answer
42 views

Find the value of this infinite product $\prod_{n=1}^{\infty }\left [ 1+\frac{1}{a_{n}} \right ]$?

Where $a_{1}=1$ and $a_{n+1}= \left ( n+1 \right )\left ( 1+a_{n} \right )$, $\forall \left ( n\epsilon \mathbb{N} \right )$ I have found that for this infinite product to be convergent, the series ...
5
votes
2answers
90 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
2
votes
2answers
55 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^{n}\frac{2k}{2k-1}$

How can I calculate the following limit: $$ \lim_{n\to \infty} \frac{2\cdot 4 \cdots (2n)}{1\cdot 3 \cdot 5 \cdots (2n-1)} $$ without using the root test or the ratio test for convergence? I have ...
0
votes
1answer
48 views

Show that the sequence of products $\prod_{k=1}^n (1+1/k^3)$ converges

$$ a_{n} = 1 + \frac{1}{n^3} $$ Show that the sequence is converges $$ \lim_{n \rightarrow \infty} \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots ...
0
votes
1answer
51 views

Is infinite product of Z a group?

Having the usual coordinate-wise addition, does infinite product of $\mathbb{Z}$ forms a group? $a,b\in \prod ^\infty \mathbb{Z}$ $a=(a_1,a_2,...)$ $b=(b_1,b_2,...)$ $a\circ ...
5
votes
2answers
101 views

Characterize the type of sequence that satisfies $\prod (1-a_i) \leq c$

Consider a product $\prod_{i=1}^{n} (1-a_i)$ where $n\leq \infty$ and $a_i\in [0,1)$ for all $i$. I'm hoping to see if there exist conditions on the sequence $\{a_i\}$ so that $$\prod_{i=1}^{n} ...
5
votes
2answers
133 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
4
votes
1answer
137 views

How find this limits $\lim_{n\to\infty}(\frac{2}{1^4}+1)(\frac{2}{2^4}+1)(\frac{2}{3^4}+1)\cdots(\frac{2}{n^4}+1)$

How to Find this limit $$\lim_{n\to\infty}\left(\dfrac{2}{1^4}+1\right)\left(\dfrac{2}{2^4}+1\right)\left(\dfrac{2}{3^4}+1\right)\cdots\left(\dfrac{2}{n^4}+1\right)$$ see 1 I ...
6
votes
3answers
76 views

Proving bounds on an infinite product

Let $p$ be an infinite product, such that $p = 2^{1/4}3^{1/9}4^{1/16}5^{1/25} ...$ Prove that $2.488472296 ≤ p ≤ 2.633367180$. I start this problem by representing p in the infinite product ...
1
vote
1answer
50 views

Prove $\prod_{n=1}^{\infty} (1+\frac{(-1)^{n+1}}{2n-1})=\sqrt2$

This question came up while I was revising the Gamma function. $$ \prod_{n=1}^{\infty} (1+\frac{(-1)^{n+1}}{2n-1})=\sqrt2 $$ Please prove me prove this infinite product, and explain the steps ...
0
votes
1answer
40 views

another infinite product problem

Is there a way to simplify the following $$ 1-\prod_{k=0}^\infty\frac{(k+a)^2-1}{(k+a)^2+b^2} $$ to a single infinite product? Assumptions: $a>0$, $b>0$. Can the following idea prove useful? $$ ...
3
votes
6answers
183 views

Proof $\frac{1}{2}\cdot\frac{3}{4}\cdot\frac {5}{6}\dots$ is zero

I would like a proof that $$ \lim\limits_{ n\to \infty }\prod_{i=1}^n\frac{2i-1}{2i}= 0 $$ It seems reasonable, although the terms approach $1$ Thank you in advance
4
votes
2answers
49 views

Solutions to $\sum{a_n} = \prod{(a_n+1)}$

$$\sum_n^\infty{a_n} = \prod_n^\infty{(a_n+1)}$$ Can you give a nontrivial example of a real sequence which satisfies this equation? By "trivial" I mean sequences such as $-1,1,0,0,0\dots$ which ...
0
votes
2answers
36 views

Simplify Product $\prod_{k=2}^{n} \left(1 - \frac{2}{k (k+1)}\right)$

I try to simplify an expression (wolfram link) but i really do not know enough about the topic. $\prod_{k=2}^{n} \left(1 - \frac{2}{k (k+1)}\right)$ I can see the result but i do not know how to ...
1
vote
2answers
37 views

How to change two elements in an uncountably infinite product

I have an uncountable product, say $$\prod_{i \in I}A_i$$ And i want to replace $A_{i_0}$ and $A_{i_1}$ by $B_{i_0}$ and $B_{i_1}$ respectively. However I know that $$\left( \prod_{i \in I, \ i ...
5
votes
0answers
97 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
0
votes
1answer
29 views

How to establish the convergence?

How to show that $\prod\limits_{n=1}^\infty \frac{1+x^{\delta^n}}{2}$ is convergent if $x>0, 0<\delta<1$ ? Thanks in advance
1
vote
2answers
30 views

How to establish the convergence of this infinite product?

How to prove that $\prod\limits_{n=1}^\infty \cos^2(\frac{1}{n^2})$ is convergent ? I have got the discussion here but could not figure out if we are asked to prove the convergence, through simple ...
2
votes
2answers
59 views

Proving $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula

I am trying to show that $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula: $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\limits_{n=1}^\infty\left(1+\frac{z}{n}\right)^{-1}e^{z/n}$$ where $\gamma$ ...
1
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0answers
31 views

Special values of the Dedekind eta function

Does anyone know of an article or a book which contains a comprehensive list of some known explicit values of the Dedekind eta function $\eta(\tau)$?
4
votes
1answer
84 views

Convergence of infinite product of prime reciprocals?

Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
3
votes
1answer
66 views

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

I'm interested in finding a closed form for $$\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$$ Wolfram Alpha confirms that it converges, but I can't find any plausible closed forms. I've made ...
1
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0answers
55 views

Infinite product: Problem I

What is the closed form value of the infinite product $$ ...
22
votes
6answers
801 views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
3
votes
1answer
72 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
111 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
63 views

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

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.
4
votes
3answers
52 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
26 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
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0answers
37 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
92 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
56 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$$\ \lim_{x\to \infty} \log \log x \prod_{i< \log x} \frac{p_i -1}{p_i}= \\ \lim_{x\to \infty} \log \log x \prod_{p < \log x}_{p prime} \frac{p-1}{p}$$?
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0answers
35 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
21 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
42 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
151 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
83 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
44 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
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0answers
38 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
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0answers
41 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
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0answers
73 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
133 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
79 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 ...
1
vote
4answers
86 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
80 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
61 views

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

How is it possible to prove the following inequality? ...
6
votes
0answers
165 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
81 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
52 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 ...