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

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Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} ...
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105 views

how to evaluate the product $\prod _{n=2}^\infty (1+ \frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots )$? [closed]

Evaluating the infinite product of $\prod _{n=2}^\infty (1+ \frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots )$. Please Help.
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37 views

nth partial product of a double product

I'm trying to find the nth partial product of: $$ \frac{5}{4}^{1/5} \prod_{m=1}^{\infty}\prod_{n=5^{m-1}}^{5^m-1}\left(\frac{5n}{5n+1}\frac{5(n+1)}{5(n+1)-1}\right)^{1/5^{m+1}} $$ I've tried ...
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How do I find the finite limits of this infinite product?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {a^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ I'd like closed form solutions, and in this case that means any ...
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1answer
47 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
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1answer
37 views

Evaluation of $\prod_{k=1}^{\infty}\frac{a+k^2}{b+k^2}$

While playing around with the question The convergence of a sequence with infinite products, I found Mathematica to give me the result $$ \prod\limits_{k=1}^{\infty}\frac{a+k^2}{b+k^2} = ...
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31 views

Growth rate of an infinite product

I have an (infinite) Blaschke product in the upper half plane, $$ f(z) = \prod_k \frac{z-z_k}{z-z_k^*}. $$ The zeros $z_k$ of the function are complex numbers in the upper half plane. Suppose the ...
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2answers
81 views

Compute the values of two infinite products whose factors are the same

I have the following question: How to prove that $(1-\frac{1}{2})\cdot (1+\frac{1}{3})\cdot (1-\frac{1}{4})\cdot (1+\frac{1}{5})\cdot (1-\frac{1}{6})\cdot (1+\frac{1}{7})\cdot ...
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1answer
48 views

Factorization $\cos(z) - \sin(z)$

How do I find the product expansion of $\cos z - \sin z$ We have $\cos z = \sin z$ iff $z = \pi/4 + k \pi$ where $k$ is an integer. The sequence $\sum (r/(|\pi/4 + k \pi|)^2$ converges For some ...
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152 views

What is the limit of this divergent infinite product multiplied by an exponential?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {c^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ My attempt: I have absolutely no clue except for the case of $c=2$ ...
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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) ...
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1answer
70 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 ...
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1answer
64 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: $$\ln(y)=\sum_{k=1}^{\infty}\ln(1-x^{-k})$$ ...
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1answer
27 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 = ...
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1answer
62 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 + ...
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144 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 ...
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1answer
53 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 ...
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0answers
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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 ...
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2answers
205 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 ...
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151 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 ...
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36 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 ...
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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 ...
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1answer
36 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 $$ ...
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53 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 $-\frac 1 {12}$. ...
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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 $a \rightarrow 1$ ...
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1answer
42 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 ...
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3answers
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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 ...
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2answers
48 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 ...
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1answer
103 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 ...
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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 ...
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1answer
74 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 ...
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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 ...
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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.
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1answer
41 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 ...
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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 ...
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2answers
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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.
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Proof of Ramanujan's Identities of Euler's Function

Consider Euler's Function defined as (and not to be confused with the totient!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
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1answer
40 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 ...
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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 ...
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1answer
50 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} ...
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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 ...
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$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$

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 ...
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1answer
32 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 ...
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1answer
24 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
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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
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1answer
88 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
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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 ...
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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? ...
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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
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1answer
282 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 ...