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

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16
votes
2answers
238 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
6
votes
1answer
90 views

Convergence of $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

I'm looking at some notes from my previous complex variables course and I need help verifying some things about the convergence of $$ \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right) $$ on compact ...
2
votes
3answers
473 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
0
votes
2answers
65 views

How to find whether this series converges or diverges?

Let's suppose I have been given a series that looks like this: $$\sum_{n=1}^n\frac{1\cdot 3\cdot 5\cdot\cdots\cdot(2n-1)}{2\cdot5\cdot8\cdot\cdots\cdot(3n-1)}$$ What I have been thinking of doing ...
0
votes
2answers
49 views

product converging to one

I have a question concerning an infinite product. Suppose $x_n$ is a sequence of positive real numbers. My intuition says that $$\lim_n(1-\exp(-x_n))^n=1$$ for any sequence $x_n=n^\alpha$ with $\alpha ...
1
vote
0answers
15 views

A sort of modified geometric series

I was wondering if there are any hints on how to manage this series $$ \sum_{i=2}^\infty \prod_{j=1}^{i-1} (1-cj^{\beta-1})=(1-c)+(1-c)(1-c2^{\beta-1})+(1-c)(1-c2^{\beta-1})(1-c3^{\beta-1})... $$ with ...
0
votes
2answers
39 views

How to do multiplication (capital pi) in WolframAlpha?

How do i ask this in WolframAlpha: $$\prod_{i=0}^{i=10} \sin{(i)}$$ I used $\text{product}(...)$ and $\text{multiply}(...)$ or even $\text{multiplication}(...)$ but they don't seem to work. I am ...
1
vote
0answers
19 views

This one weird infite product can define exponentials in terms of itself. What does it do for other constants?

What is... $$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ This is similar to my other question. However, there is a constant factor rather than variable in the ...
4
votes
3answers
74 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
2
votes
2answers
29 views

Ratio test for infinite products?

Is the ratio test applicable for testing convergence of infinite products? In other words, consider the sequence $(a_i)_{i=1}^\infty$ of non-zero real numbers. Also, consider the product ...
5
votes
3answers
145 views

An infinite product for $e^{x}$

This is again from MathWorld. Let $ x > 0$ and define a sequence $a_{n}$ recursively as follows $$a_{1} = \frac{1}{x}, a_{n} = n(a_{n - 1} + 1)$$ Show that $$e^{x} = \prod_{n = 1}^{\infty}\left(1 + ...
0
votes
0answers
16 views

Prove that $\lim_{R \to \infty} \int_{\gamma_R} \frac{z}{(z+2)^3} \prod_{n=1}^{\infty} \frac{\lambda_n - z}{2+\lambda_n+z} dz = 0$

...where $\gamma_R$ is the semicircle in the half plane $\{z \in \mathbb{C}:\Re(z) > -1\}$ of radius $R$ centered at $-1$, assuming that $\sum_{n=1}^{\infty} 1/\lambda_n < \infty$, ...
3
votes
0answers
37 views

Partitions of integers, a series for infinite product $(1+q)(1+q^3)(1+q^5)\cdots$

Show that $$ (1+q)(1+q^3)(1+q^5) \cdots = 1+ \sum_{k=1}^\infty \frac{q^{k^2}}{(1-q^2)(1-q^4)(1-q^6) \cdots (1-q^{2k})}.$$ How would one proceed combinatorially. What I know is that the left-hand ...
1
vote
0answers
25 views

Uniform convergence of an infinite product of automorphisms of the unit disk.

I need to show that $\Pi \frac{|a_n|}{a_n} \frac{a_n - z}{1-\overline{a_n}z}$ with $\{a_n\} \subset D(0,1)$ and $\Sigma$ $(1 -|a_n|) < \infty$ converges to a holomorphic function in $D(0,1)$ It's ...
1
vote
1answer
29 views

Show that $(1+z)\Pi^{\infty}(1+z^{2^{n}})=\frac{1}{1-z}$ for $z \in \mathbb{C}$ $|z|<1$

$(1+z)\Pi^{\infty}(1+z^{2^{n}})=\frac{1}{1-z}$ for $z \in \mathbb{C}$ $|z|<1$. Any tricks to prove this? I'm not really sure how to start.
2
votes
2answers
121 views

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} ...
4
votes
3answers
136 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.
0
votes
0answers
42 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 ...
4
votes
0answers
36 views

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 ...
3
votes
1answer
50 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 ...
1
vote
1answer
42 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} = ...
3
votes
2answers
83 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 ...
0
votes
1answer
50 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 ...
7
votes
1answer
169 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 $c=2$ ...
4
votes
0answers
66 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
75 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
65 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
32 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
72 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
148 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
55 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
38 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
210 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
160 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
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 ...
3
votes
0answers
131 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
42 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
59 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
39 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
44 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
52 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
106 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 ...
6
votes
2answers
84 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
79 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 ...
8
votes
2answers
106 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
96 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
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 ...
3
votes
2answers
39 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
1k 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.