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

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4
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0answers
46 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
5
votes
1answer
68 views

Solving $y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}$

There is the trivial $y=0$, but beyond that, could there be further solutions for $y$ in terms of $x$ such that $$y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}\mbox{ pointwise}$$ ? I posed this problem ...
0
votes
1answer
69 views

Need help with calculating this product: [closed]

I need help with calculating this product: $$\prod_{n=2}^{\infty}\frac{n^5-1}{n^5+1}$$
0
votes
1answer
62 views

Maximise $y$ with respect to $x$ for $y=\prod_{k=1}^{\infty}(1-x^{-k})$

$$y=\prod_{k=1}^{\infty}(1-x^{-k})$$ I want to maximise this function. So far I have: $$\ln(y)=\sum_{k=1}^{\infty}\ln(1-x^{-k})$$ ...
0
votes
1answer
24 views

convergence of an infinite product (how to prove ?)

Fix $C > 0 $ a constant and fix $n \in N$. Consider $\alpha \in (0,1)$ fixed. I am reading a paper and the authors says: For an arbitrary $r <1$ , the infinite product $$P = ...
2
votes
1answer
51 views

Calculation of $\prod_{k=1}^\infty \left( 1 + \frac{a}{k^2} \right)$? [duplicate]

I am curious how to calculate the infinite product $$ \prod_{k=1}^\infty \left( 1 + \frac{a}{k^2} \right). $$ WolframAlpha reports that it is equal to approximately $$ \prod_{k=1}^\infty \left( 1 + ...
4
votes
3answers
138 views

To prove $\prod\limits_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$

Prove $$\prod_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$$ This equation may be famous, but I have no idea how to start. I suppose it is related to another eqution: (Euler)And how can I ...
3
votes
1answer
49 views

How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?

I would like to test whether or not the following product (or its complement) $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$ converges ...
1
vote
0answers
37 views

Product over real interval? Is there a better way of putting this?

In my amateur interest, I have arrived at this (nothing rigorous here at all):$$\prod_{a\in [1,2]} \prod_{b=0}^\infty f(a,b) \neq 0$$ For starters, there might be a more intuitive way about doing ...
2
votes
2answers
204 views

Infinite Product - Seems to telescope

Evaluate $$\left(1 + \frac{2}{3+1}\right)\left(1 + \frac{2}{3^2 + 1}\right)\left(1 + \frac{2}{3^3 + 1}\right)\cdots$$ It looks like this product telescopes: the denominators cancel out (except the ...
5
votes
1answer
143 views

Power series expansion of Blaschke product

Suppose $B$ is a Blaschke product with at least one zero off the origin, and $B(z)=\sum_{k=0}^\infty {c_kz^k}$. Is it possible that $c_k\ge0$ for all $k=0,1,\ldots$? My try: Since $B(z)$ takes real ...
1
vote
0answers
33 views

Rate of Convergence of $A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$

I'd like to know how fast the infinite product $$A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$$ converges, where the product is taken ...
1
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0answers
29 views

If $\alpha = \prod_{i = 1}^{\infty} a_i \in \bar{\Bbb{Q}}$ can we write $\alpha = a_n \alpha'$ with $\text{den}(\alpha) = \text{den}(\alpha')$?

If $\alpha$ is any algebraic number, there is an integer $d > 0$ such that $d\alpha$ is an algebraic integer, and the minimum such $d$ is called the denominator of $\alpha$, written ...
1
vote
1answer
34 views

Is there a limiting case for this sequence of infinite product representations for the theta function?

Starting from the famous infinite product $$ (1+z)^2(1-z^2)(1+z^3)^2(1-z^4)(1+z^5)^2(1-z^6)\cdots=1+2z+2z^4+2z^9+2z^{16}+\dots $$ it is easy to show by induction that $$ ...
0
votes
0answers
51 views

A way to sum supernatural numbers involving Zeta function's analytic continuation

I have this idea on how to sum supernatural numbers assigning them a finite value in a way similar to how we assume that the sum of every natural numbers from 1 to infinity equals $-\frac 1 {12}$. ...
1
vote
0answers
27 views

Q Pochammer Symbol Product Identities

Consider the expression $$G(x,a) = \frac{1}{((1-a)x;a)_{\infty}}$$ Based on: Infinite sum involving ascending powers It follows that in the limit as $a \rightarrow 1$ ...
2
votes
1answer
40 views

Divergence of $\prod_{n=2}^\infty(1+(-1)^n/\sqrt n)$.

Looking looking for a verification of my proof that the above product diverges. $$\begin{align} \prod_{n=2}^\infty\left(1+\frac{(-1)^n}{\sqrt n}\right) & =\prod_{n=1}^\infty\left(1+\frac1{\sqrt ...
7
votes
3answers
1k views

How do I find the value of this weird expression?

How can I find the value of the expression $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^...}} $? I wrote a computer program to calculate the value, and the result comes out to be 2 (more precisely 1.999997). Can ...
1
vote
2answers
47 views

Generating function for partitions

It is a theorem of Euler that $$\sum p(k)x^k=\prod\frac{1}{1-x^k}.$$ Something which annoys me is how to interpret the right hand side. I know that one can do this analytically, but I would like a ...
6
votes
1answer
102 views

What does $\Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ do?

Take the product of rings $M = \Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ over the primes or in general take any infinite set of quotient modules of a ring $R$ and form their product. It's true ...
5
votes
2answers
73 views

What is the value of $\prod_{i=1}^\infty 1-\frac{1}{2^i}$?

Also, what about in general, for some value p, which has the value 2 in the given formula? MOTIVATION: I was wondering the ...
0
votes
1answer
68 views

Infinite product converges to meromorphic function

How do you show that $\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is ...
7
votes
2answers
91 views

Anybody know a proof of $\prod_{n=1}^\infty\cos(x/2^n)=\sin x/x$.

This is actually an exercise from Apostol's Mathematical Analysis. Ch. 8 Ex 42. which asks to find all real values $x$ for which $\prod_{n=1}^\infty \cos(x/2^n)$ converges. I've shown that the product ...
5
votes
3answers
93 views

How to prove that $\prod_{n=0}^\infty \frac{(4n+2)^2}{(4n+1)(4n+3)}=\sqrt{2}$

How to prove that $$\displaystyle\prod_{n=0}^\infty \frac{(4n+2)^2}{(4n+1)(4n+3)}=\sqrt{2}$$ Thanks in advances.
2
votes
1answer
39 views

Interchanging summands among infinitely many infinite series

I am reading the following lecture notes concerning analytic number theory: http://www.math.uiuc.edu/~hildebr/ant/main4.pdf On the pages 111/112 the partial product $P_N(s)$ is defined. Then some ...
3
votes
2answers
36 views

Is it possible to write Catalan's product for e as a product of a sequence?

Catalan found a product for $e$: $$e=\dfrac{2}{1}\left(\dfrac{4}{3}\right)^{\frac{1}{2}}\left(\dfrac{6\cdot 8}{5\cdot 7}\right)^{\frac{1}{4}}\left(\dfrac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot ...
8
votes
2answers
993 views

An infinite product

I am trying to compute the infinite product $$ \prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right) . $$ Wolfram Alpha says the result is $2$, but I can't seem to figure out why.
0
votes
0answers
43 views

Proof of Ramanujan's Identities of Euler's Function

Consider Euler's Function defined as (and not to be confused with the totient!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
2
votes
1answer
38 views

How to Show $\prod_{n=1}^\infty\bigg(\frac{z^n}{n!}+e^{\frac{z}{2^n}}\bigg)$ Converges Uniformly on Compact Sets

Prove that $$\prod_{n=1}^\infty\bigg(\frac{z^n}{n!}+e^{\frac{z}{2^n}}\bigg)\tag{$*$}$$ converges uniformly on compact sets to an entire function. I haven't seen a problem like this before, so I ...
2
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0answers
26 views

Question concerning the influence of interchanging factors of an infinite product on the value of the product

I am searching for a proof of the following fact: If an infinite product $\prod\limits_{n=1}^{\infty} (1+a_n)$ of complex numbers is absolutely convergent, then its factors can be reordered ...
1
vote
1answer
46 views

Question concerning the conditional convergence of some infinite products

Let ${(a_n)}_{n=1}^{\infty}$ be a sequence of complex numbers. Let $\sigma:\mathbb{N}\rightarrow \mathbb{N}$ be a bijective map. Is it possible then that $\prod\limits_{n=1}^{\infty} ...
2
votes
2answers
72 views

Is it possible to turn infinite sums into infinite products?

I am working on studying infinite products. I know that it is possible to convert an infinite product to an infinite sum using logarithms, where $$\log \prod s_n = \sum \log s_n$$ My question is, it ...
2
votes
3answers
82 views

$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$

Let $a_i \ge 0$ $$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$$ I've got to this step $$\prod_{i=1}^\infty (1+a_i) = e^{\sum_{i=1}^\infty ...
0
votes
1answer
31 views

Products of torsion groups

Given an infinite family of non-zero torsion groups $G_i$ (not necessarily commutative). Prove that their Cartesian product is a torsion group iff all but finitely many (i.e. "almost all") of the ...
0
votes
1answer
23 views

What's a “Basis of Measurable Sets?”

As defined here http://modular.math.washington.edu/129/ant/html/node82.html Using the notation in the link, one takes sets of the form $\prod\limits_{\lambda} M_{\lambda}$, where each $M_{\lambda}$ ...
2
votes
1answer
62 views

What is the value of this Infinite Product of prime numbers expression? [duplicate]

What is the value of: $$\prod_1^\infty \frac{p_i^2}{p_i^2 -1 }$$ Where $$p_i$$ are the prime numbers: 2, 3, ...
2
votes
1answer
81 views

Order of an entire function represented as an infinite product

Question- What is the order of growth of the entire function given by the infinite product of $1-(z/n!)$ where $n$ goes from 1 to infinity? My thoughts- I have already proven that the infinite ...
2
votes
2answers
74 views

Non vanishing of an infinite product

I need to prove that the infinite product $$\prod_n \left(1-\frac{1} {(a^n+1)^2} \right)^{\frac{a^n}{n}} $$ with $a$ an integer $\geq 3$, converges to a real number $L$ such that $0<L<1$. It's ...
1
vote
2answers
100 views

The limit of product $\prod_{x=2}^k(1-x^{-2})$ as $k\to\infty$

For some reason Wolfram is saying that as $k$ tends to infinity, $\prod_{x=2}^k(1-x^{-2})$ tends to zero, but my book is claiming that this product is never less than one half. Which is true, and why? ...
0
votes
1answer
36 views

How to prove that if $\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$ then, for any $0<c_k<b_k$, $\lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l$?

How to prove that if $$\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$$ then for any $0<c_k<b_k$ $$ \ \ \ \ \ \ \ \ \ \ \lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l \ \ \ \ \ ?$$ ...
2
votes
1answer
208 views

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 ...
3
votes
5answers
106 views

Evaluation of $\lim_{x \to k\pi} \frac{\sin(x)}{x(x-k\pi)}$ from the Weierstrass product expansion of $\sin(x)$

Consider the Weierstrass infinite product expansion of the $\sin(x)$ function: $$ \tag 1 \frac{\sin(x)}{x} = \prod_{n=0}^\infty \left( 1 - \frac{x^2}{n^2 \pi^2} \right) = \prod_{n\neq 0} \left( 1 - ...
1
vote
0answers
25 views

Find the values of x for which $\sum_{n=0}^\infty (-\frac{1}{2})^n(x-3)^n$ converges

I am given a geometric series $\sum_{n=0}^\infty (-\frac{1}{2})^n(x-3)^n$ and am asked to find the values of x for which the given geometric series converges then find the sum of the series (as a ...
2
votes
2answers
147 views

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 ...
2
votes
0answers
39 views

$\lim_{n \to \infty}\prod_{k=1}^n \cos(k\sqrt{\frac{3}{n^3}} t) = e^{- \frac{t^2}{2}}$ [duplicate]

Is it true that $$\lim_{n \to \infty}\prod_{k=1}^n \cos\left(k\sqrt{\frac{3}{n^3}} t\right) = e^{- \frac{t^2}{2}}$$ ? How to proceed ?
3
votes
1answer
93 views

show that $e=(\frac{2}{1})^{\frac{1}{1}}(\frac{4}{3})^{\frac{1}{2}}(\frac{6\cdot8}{5\cdot7})^{\frac{1}{4}}…$

I found $$e=(\frac{2}{1})^{\frac{1}{1}}(\frac{4}{3})^{\frac{1}{2}}(\frac{6\cdot8}{5\cdot7})^{\frac{1}{4}}(\frac{10\cdot12\cdot14\cdot16}{9\cdot11\cdot13\cdot15})^{\frac{1}{8}}.....$$ easily you can ...
0
votes
0answers
51 views

a question concerning infinite product

I have a question about the convergence of infinite product. In real mathematics analysis page 198 question 63, part a) $a_k$=$(-1)^k$/${\sqrt{k}}$, and we need to show series ...
0
votes
3answers
112 views

convergence of $(1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$

I was given a problem of testing sequence convergence. The sequence is defined as: $$x_n= (1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$$ My first idea was to define $y_n$ as follows: $$y_n = ...
0
votes
1answer
46 views

How can I find the sum of an infinite series of products?

Background Today in my macroeconomics class my teacher taught us three concepts. The first is very simple: consumption $c$ is a linear function of national income $y$. Mathematically, $$c = My + ...
-2
votes
2answers
206 views

Inequality $\prod\limits_{r=1}^{\infty}(1+(\frac{1}{2})^r)<\frac 52$ [closed]

Prove this inequality. $\prod\limits_{r=1}^{\infty}\left(1+\left(\frac{1}{2}\right)^r\right)<\dfrac 52$ I have tried to prove it using induction but it is not coming.