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

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3
votes
1answer
54 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 ...
4
votes
2answers
121 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 ...
0
votes
0answers
35 views

NonCountable Product of Discrete Space {0,1}

Is the product of $\{0,1\}^I$ for any $I$ metrizable (with the product topology)? It Would be helpful to see proof\disproof idea. Thank you
2
votes
1answer
52 views

Why does this product diverge?

Consider the partial product $$ p_n=\prod_{k=1}^{2n}a_k,$$ where $$a_k=\left\{\begin{array}{ll} k & \text{for }k\text{ odd} \\ \frac{1}{k-1} & \text{for }k \text{ even} \end{array}\right..$$ ...
4
votes
0answers
56 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
55 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
43 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
59 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
80 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
17 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
1answer
26 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
99 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
43 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
110 views

Find the product of the series? [closed]

Can anybody please help me to evaluate the following product? $$\prod_{n=2}^\infty\dfrac{n^3 - 1}{n^3 + 1} $$
3
votes
1answer
80 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
66 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
85 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 ...
3
votes
2answers
111 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
42 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 ...
2
votes
2answers
95 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
92 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
49 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
63 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
55 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
46 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
59 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 ...
1
vote
0answers
54 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
1
vote
1answer
38 views

$\frac{f'(z)}{f(z)}= \sum_{n=1}^{+ \infty}\frac{f'_{n}(z)}{f_{n}(z)}$

I 've found this exercise. Let $\{f_{n}\}$ be a sequence of holomorphic functions on a given domain $\Omega$. Suppose that $\prod_{1}^{\infty}f_{n}$ converges uniformly on compact subsets of $\Omega$ ...
3
votes
3answers
91 views

Finding limit of a sequence in product form

\begin{equation} \prod_{n=2}^{\infty} \left (1-\frac{2}{n(n+1)} \right )^2 \end{equation} I need to find limit for the following product..answer is $\frac{1}{9}$. I have tried cancelling out but ...
1
vote
1answer
34 views

Convergence condition of infinite cosine product

Please show that, given that $\sum_{k\ge1}c_k^2=\infty$ and $c_k\rightarrow 0$, $$\lim_{n\rightarrow\infty}\prod_{k=1}^n\cos{tc_k}=0$$ for every $t\neq0$. (All variables here are real numbers.) The ...
10
votes
3answers
182 views

Computing $\lim_{n\to \infty}{\frac{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)}{7\cdot11\cdot15\cdot\dots.\cdot(4n+3)}}$

Let $\{a_n\}_{n\ge1}^{\infty}=\bigg\{\cfrac{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)}{7\cdot11\cdot15\cdot\dots.\cdot(4n+3)}\bigg\}$. Find $\lim_{n\to \infty}{a_n}$. I.) In the first step I studied ...
3
votes
1answer
68 views

Find $\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$

Find the following limit: $$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$$ The numerator can be simplified by using Euler's formula and the sum of ...
1
vote
0answers
22 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
6
votes
2answers
67 views

Find $\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$

Evaluate the following integral $$\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$$ I was thinking of a way which do not need to explicitly find the closed form of the infinite product, ...
0
votes
0answers
31 views

Is the explicit formula for the second chebyshev function unique?

Is the explicit formula for the second chebyshev function unique ? Or is it possible there are multiple explicit formula ? Are there explicit formula's given as an infinite product over the zero's ...
6
votes
1answer
95 views

Is there an infinite group that contains every finite group (and no infinite group) as a subgroup?

Question is in the title. For bonus points, construct the group $G$ such that it also has no infinite proper subgroups. (This second question relates to the Prüfer group, but that group is abelian, ...
2
votes
0answers
38 views

Product Involving Sines

I'm studying the following product: $$p(a,\omega)=\prod_{k=1}^{\infty}a\sin (k\omega\pi),\quad \omega \in \Bbb R,\quad a\in \Bbb R_+.$$ It's easy to see that for $a\in (0,1]$ this product diverges to ...
2
votes
1answer
67 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
2
votes
2answers
61 views

Infinite product $(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})$

I have to show that if $|z| < 1$, $z \in \mathbb{C}$, $$(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})= \frac{1}{1-z}$$ I want to understand how to do this kind of exercises, any hint ?
0
votes
2answers
51 views

Solving infinite sums with primes.

Let $p_n$ denote the $n$'th prime number. How would one go about proving that infinite products like: $$\prod_{k=1}^\infty1 - \frac{1}{(p_k)^2} = \frac{6}{\pi^2}$$ or ...
1
vote
2answers
66 views

Infinite product convergence

Prove that $$\prod_n\left(1+\frac{i}{n}\right)$$ diverges. But $$\prod_n\left\vert 1+\frac{i}{n}\right\vert$$ converges. I know the theorem $\prod (1+z_k) $ converges $\iff$ $\sum\log (1+z_k)$ ...
5
votes
3answers
182 views

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
vote
0answers
54 views

Infinite sum of infinite products

Is there a solution for the infinite sum of the following form?? $x^y + x^{y + y^2} + x^{y + y^2 + y^3} + x^{y + y^2 + y^3 + y^4} + ...$ It can also be represented this way: $\sum_{j = 1}^{\infty} ...
4
votes
2answers
361 views

Evaluating an Infinite Product

Does anyone know how to evaluate the infinite product $$ \left(1 - \frac{4}{1}\right) \prod_{k = 3}^{\infty} \left( 1 - \frac{4}{k^2} \right) $$
3
votes
5answers
177 views

Product of all primes

Is the product of all primes a natural number? In other words, is this true: $$ \prod\limits_{\text{primes}} p_i \in \mathbb{N} $$ And if so, what about just some of them: $$ ...
4
votes
2answers
85 views

A curious product formula

Fiddling with Mathematica seems to suggest the following: $$\frac{(2^2)(4^2)(6^2)\cdots(2N^2)}{(1^2)(3^2)(5^2)\cdots(2N-1)^2}=N\pi+\frac{\pi}{4}+\frac{\pi}{32N}-\frac{\pi}{128N^2}+o(1/N^2).$$ Does ...
1
vote
0answers
29 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
2
votes
1answer
50 views

Choosing a branch for $\log$ when comparing $\prod_{n=1}^\infty(1+a_n)$ and $\sum_{n=1}^\infty \log{(1+a_n)}$

On Ahlfors on p. 191 he is talking about the relation between $\prod_{n=1}^\infty (1+a_n)$ and $\sum_{n=1}^\infty \log(1+a_n)$. He says: Since the $a_n$ are complex, we must agree on a definite ...
2
votes
1answer
344 views

What is $\lim\limits_{n\to\infty}(\sqrt2-\sqrt[3]2)\cdots(\sqrt2-\sqrt[n]2)$? How to approach? [duplicate]

$$\lim\limits_{n\to\infty}(\sqrt2-\sqrt[3]2)(\sqrt2-\sqrt[4]2)(\sqrt2-\sqrt[5]2)\cdots(\sqrt2-\sqrt[n]2)$$ Could you tell me how to approach this kind of question? How do I find the limit of this ...
4
votes
1answer
53 views

Show $g(z+1) = zg(z)$

This is for homework, and I am in need of a hint. Given the product $$ g(z) = \prod_{k=1}^{\infty} \frac{k}{z+k}\left( 1 + \frac{1}{k} \right)^z, $$ I am trying to show that $g(z+1) = zg(z)$. Here ...