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Questions tagged [infinite-product]

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

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2answers
56 views

General closed form for $L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx$ when $\phi\in(0,\pi)$?

I would like to know if there is a general closed form for $$L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx,\qquad \phi \in(0,\pi)$$ Context: (below are also the extent of my search for a closed form....
4
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1answer
52 views

Convergence of $\sum\limits_{n=0}^{\infty} (1-|a_n|)$

So I must prove that if $(a_n)$ is a sequence of points in $\mathbb{C}$ with $0< |a_n| < 1 \; \forall n \in \mathbb{N}$ and verifying that $|b| \leq \prod\limits_{n=1}^{\infty} |a_n|$ with $0&...
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0answers
41 views

Monotonicity of an infinite product

It holds that $f(k,\lambda):=(k^2-\lambda)\prod\limits_{l \in \mathbb{Z}\setminus\{0\}} \frac{(k+l)^2-\lambda}{l^2}=\frac{1}{\pi^2} \sin(\pi(k+\sqrt{\lambda}))\sin(\pi(k-\sqrt{\lambda}))$ by the ...
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1answer
12 views

Is the following statement about matrix norms of matrix products correct?

Let $A$ be a real-valued, square matrix and define its 2-norm as: $$||A||_2 = \sqrt{\max_i\lambda_i(AA^T)}$$ where $\lambda_i(AA^T)$ denotes the $i^{th}$ eigenvalue of the product $AA^T$. Now ...
2
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1answer
43 views

Simplifying a product of a series

I tried simplifying the product $$\prod_{k=1}^{\infty}\left[1-x^k\right]$$ by factoring it into $$\prod_{k=1}^{\infty}\left[\left(1-x\right)\sum_{i=0}^{k-1}x^i\right].$$ I am not very experienced in ...
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0answers
24 views

Infinite products of Laurent series

I am trying to find an expression for the coefficients of a Laurent series which is itself an infinite product of Laurent series: $f(z) = \sum_{u=-\infty}^{\infty} f_{u}z^{u} = \prod_{i=0}^{\infty} \...
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70 views

I need a help with my research. Just asking for inventive answer to my question. [on hold]

Let's take a value and give it to x. Let's say x has its growth doesn't matter how much. Let's describe his growth on a number scale like a man on a boat going in a direction of a river flow. Let's ...
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0answers
20 views

Do the Airy Functions have a product representation?

I was wondering if the Airy Functions $$Ai(z)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(zt+\frac{t^3}{3})}dt$$ $$Bi(z)=\frac{1}{\pi}\int_0^\infty e^{-\frac{t^3}{3}+tz}+\sin\left(\frac{t^3}{3}+zt\right)...
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1answer
20 views

Why can('t) I map this closed subset of a product of compact sets to a non-compact set?

Let $S$ be a countable set of real numbers that is bounded but has neither a maximum nor a minimum. Next we create the product topology $[0, 1]^S$ (using the usual topology on $[0, 1]$). This should ...
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2answers
141 views

How to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$?

I need to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$ but wolframalpha's brain is too narrow to contain the result, and I don't know any software and programming to find ...
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0answers
14 views

Methods for assigning values to infinite constructs (ex. sums, products)

(This question is more to get an overhead view of the topic rather than a well defined answer) How can one assign values to infinite constructs AND justify them. What do i mean with infinite ...
4
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1answer
88 views

Evaluate $\frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}\cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots$

Problem Evaluate the infinite product $$\frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}\cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots$$ Attempt For convenience,let's rewrite the limit. ...
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2answers
216 views

When does the tensor product distribute over an infinite direct product?

It is well known that the tensor product of $R$-modules over some ring $R$ does not, in general, distribute over infinite direct products, an obvious example being $\mathbb Z_p \otimes_\mathbb Z \...
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2answers
46 views

Demonstrating an equivalent formula for $\sin(x)/x$ using power series

Would appreciate some ideas for the following: "Prove that $\frac{\sin{x}}{x}=\prod_{n=1}^{\infty}\cos{\frac{x}{2^n}}$ using power series." I'm aware this identity can be shown using trig identities ...
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1answer
27 views

Verifying that $ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \frac{1}{(1-q^{2j-1})(1-q^{2j})}$

On page 165 of Chapter 13, how was the equality made from line 1 to line 2? https://archive.org/details/NumberTheory_862/page/n173 Namely, how $$ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\...
26
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1answer
711 views

Integral $\int_0^1 \frac{dx}{\prod_{n=1}^\infty (1+x^n)}$

The following integral appeared this summer on AoPS. However it received no answer until today. $$I=\lim_{n\to \infty } \int_0^1\frac{dx}{(1+x)(1+x^2)\dots(1+x^n)}=\int_0^1 \frac{dx}{\prod_{n=1}^\...
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2answers
45 views

How can I show that this infinite product is nonzero?

How would you show that $\prod_{k=1}^\infty \cos( 2 \pi/3^k)$ is nonzero? Wolfram approximates it as about $-0.37$, and I have a guess that $$ \Big \vert \prod_{k=1}^\infty \cos( 2 \pi/3^k) \Big \...
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1answer
50 views

Does the infinite product of the reciprocals of a decreasing (or increasing) function equal zero? [closed]

I've made a short document explaining what I've just claimed. I'd like to know if the criteria for the infinite product to be zero is enough to hold for all decreasing and increasing functions. ...
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0answers
186 views

If a product of polynomials converges, does some product of their zeros also converge?

Suppose $\{p_k\}$ is a sequence of polynomials with $p_k(0)=1$. Let $a_1,a_2,\ldots$ be an enumeration of all of the zeros of the $p_k$. Suppose that $$\prod_{k=1}^\infty p_k(z)$$ converges ...
7
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3answers
109 views

Prove that $\prod_{i=1}^n (1+x_i/n) \sim \exp (\sum_{i=1}^n x_i/n)$ as $n\rightarrow\infty$?

Let $x_1,x_2,\dots$ be an infinite sequence of real numbers. Assume that they are bounded, $|x_i| \le C < \infty$ for all $i$ for some $C$. Is it true that, for any such sequence $$\lim_{n \...
9
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2answers
141 views

Show that $\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$ [duplicate]

$$\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$$ This monstrous expression is from Tolaso Network (tolaso.com.gr). ...
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1answer
64 views

Is this zeta-type function meromorphic?

In An older question I asked : ( See A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH) ) —— Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ ...
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0answers
39 views

Need someone to show me how the Zeta function is equal to Euler's product formula?

$\zeta(x)$=$1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\frac{1}{5^x}+\frac{1}{6^x}+\cdots$Euler's product formula states, for all $x$ greater than $1$ we have:$\zeta(x)$=$1\over 1-\frac{1}{2^x}$$\...
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3answers
70 views

Solution to infinite product $\prod_{p-primes}^{\infty} \frac{p}{p-1}$

I want to find the $\prod_{p-primes}^{\infty} \frac{p}{p-1}$. This question stems from a question from amc 12a 2018. It goes as follows: Let $A$ be the set of positive integers that have no prime ...
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0answers
19 views

What is the terminology for a subset of a product of sets that is the product of its cross-sections?

Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind ...
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1answer
34 views

Convergence to zero of a specific infinite product

Maybe less ambitious but I would be happy to prove the following. I have tried but without success. Thanks for any help
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0answers
39 views

Particular infinite product convergence

My question is a little bit technical but if someone has a clue... I have the following infinite product $$ P = \prod_{n=1}^\infty( 1 - q^n(n)) $$ where $q(n)$ is an increasing sequence whose limit ...
2
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1answer
28 views

Sequence from a generating function?

Find the sequence $a_n$ if its generating function is $$A(x) = \prod_{n=1}^{\infty}(1-q^nx)$$ Well, i need to find the expansion of A(x) in terms of powers of x. For that I take the log of both the ...
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3answers
83 views

Calculate an infinite series by computer: get sum and number of terms for given precision

I need to find a sum and number of terms in an infinite series if ε ∈ (0;1) and x ∈ (1;5): $$\sum_{k=0}^∞ \frac{k^2x^k}{(k+1)!}$$ I was able to convert it into a simpler form, but stuck there: $$\...
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2answers
76 views

Question on uniform convergence of sum of continuous functions.

i) I have been stuck on this for quite some time. Can anyone explain how $h(x)$ converges uniformly(and absolutely) given the inequality. I don't think I can use Weierstrass M-Test. ii) Secondly, ...
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0answers
32 views

Find the infinite product series [duplicate]

$ \lim_{n\to\infty} (1-\frac{1}{2})(1-\frac{1}{4}) \cdots (1-\frac{1}{2^n})=\prod_{n=1}^{\infty} (1-\frac{1}{2^n}) $ I've tried this techniques: 1) Using squeeze theorem, but i had not find right ...
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1answer
28 views

Metrizability of $\Bbb{R}^I$ and the Sequence Lemma

Let $I$ be some uncountable set, and let $\Bbb{R}^I$ denote the product of uncountably many copies of $\Bbb{R}$. Show that $\Bbb{R}^I$ is not metrizable. I know that there is a solution which ...
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85 views

Integrating $\frac{x}{\sin x}$ using infinite products and fraction decomposition

I found this way of integrating $\frac{x}{\sin x}$ using infinite products and fraction decomposition. $$I=\int\frac{xdx}{\sin x}=\int\frac{xdx}{x\prod_{n\geq1}(1-\frac{x^2}{\pi^2n^2})}\\I=\int\prod_{...
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1answer
53 views

limit of product

Let $k=n/m \to \infty$ as $n \to \infty$. I am wondering whether we have $$\lim_{n\to \infty}\prod_{i=1}^{k}\left(1-\frac{i}{n}\right)=1?$$ I can prove that the limit above exists and equals $1$ ...
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0answers
77 views

Does $\prod_{k=3}^{\infty}\cos\frac{2\pi}{n!}$ have a closed-form solution?

How to compute this please? $$\prod_{k=3}^{\infty}\cos\frac{2\pi}{k!}$$ I just know it is convergent.
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8answers
290 views

Show that $\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\cdots\left(1+\frac{1}{n^3}\right) < 3$

I have this problem which says that for any positive integer $n$, $n \neq 0$ the following inequality is true: $$\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\...
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2answers
30 views

Closed form of a product of a ratio of polynomials

I have been trying to approach finding a closed form of the following product$$\prod_{n=0}^\infty\frac{n^2+x_{1}n+x_{0}}{n^2+y_{1}n+y_{0}}$$ and in general$$\prod_{n=0}^\infty\frac{n^m+\sum_{k=0}^{m-...
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1answer
80 views

Convergence of infinite product $\prod(1+a_n)$ where $a_n$ changes sign

I know that an infinite product $\prod_{n=1}^\infty (1+a_n)$ with $a_n \geq0$ for all $n$ converges if and only if the series $\sum_{n=1}^\infty a_n$ converges. I can prove this using the inequality $...
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3answers
84 views

Prove that $\prod\limits_{k=1}^\infty \left(1+\frac1{2^k}\right) \lt e ?$

How would you prove that $$\displaystyle \prod_{k=1}^\infty \left(1+\dfrac{1}{2^k}\right) \lt e ?$$ Wolfram|Alpha shows that the product evaluates to $2.384231 \dots$ but is there a nice way to write ...
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1answer
21 views

How do I interpret *all factors have a common finite exponent* in this context?

"The product of infinitely many torsion groups will no longer be a torsion group unless all factors have a common finite exponent (which is not the case if we take Prüfer groups)." How do I interpret ...
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2answers
107 views

Is this sum constant for n?

Hi I can prove that this sum is constant in $n\in \mathbb{N}$. However my proof is very long (a few pages with probability involved). Does anyone see a simple proof. The sum in question is (a q-series ...
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2answers
22 views

Is the sequence of real number that are $0$ from some point Sense in the box or product topologies

Let $A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$, series of real numbers that are zero from some point forward. ...
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0answers
11 views

The existence of a moment for the product of an ergodic sequence of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of bounded stationary ergodic random variables with $$ \log(X_1) <0. $$ Then it can be easily shown, using Birkoff's ergodic theorem, that almost surely $$...
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1answer
53 views

Why is the proof of “countable many product of countable sets is countable” wrong? [duplicate]

As this question clearly shows that the countable many product of countable sets is uncountable. However, I do not understand why the below proof is wrong: (False) Proof: Let $A$ be a countable set. ...
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3answers
61 views

Convergence of $P=\prod_{k=1}^{\infty}\left(1+\frac{1}{5^k}\right) $

Convergence of $$P=\prod_{k=1}^{\infty}\left(1+\frac{1}{5^k}\right) $$ Will this product converges to finite limit? My try: we have $$P=1+\frac{1}{5}+\frac{1}{5^2}+2\frac{1}{5^3}+\cdots+2\frac{1}{5^...
0
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0answers
43 views

Infinite sum $S(k) = \lambda + 2^k \lambda^2 + 3^k \lambda^3 + \dots$

I'm trying to find an expression for: $$S(k) = \sum_{n=0}^{\infty} n^k \lambda^n$$ I have found a recursive expression for this where I first find $S(0)$, then use that to find $S(1)$. Then use those ...
0
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0answers
25 views

Is an infinite direct sum of modules still a product? Coproduct?

I know nothing about categories, but today I did an exercise proving that for a module $M$ : $M, \xi_1,\dots, \xi_d$ is a product $\iff M \cong \bigoplus_{i=1}^d M_i \iff$ $M, \alpha_1,\dots,\alpha_d$...
2
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2answers
61 views

On $\sin x=x \cdot \prod_{n=1}^{\infty}(1-(\frac{x}{\pi n})^2)$

I found this product representation of $\sin x$ online and I want to know the intuition behind this. I know that there are other questions about this product in this site, but I haven't gotten what I ...
2
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3answers
55 views

Why is $\prod_{n=1}^\infty \dfrac{(1+1/n)^s}{1+s/n}$ convergent for $s$ complex with $Re(s)>0$?

Let $s$ be a complex number with $Re(s)>0$. I am trying to show that $$f(s) = \prod_{n=1}^\infty \frac{(1+1/n)^s}{1+s/n}$$ converges. My approach was to look at the sum $$\sum |1-f(s)|$$ ...
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0answers
58 views

Closed form of this product or approximate?

What is the closed form of this product: $$\prod_{i=1}^{k-1}\left(1-e^{-a(b- ic)^2}\right)$$ where $a,b,c$ are constants?