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Questions tagged [infinite-product]

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

0
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2answers
41 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 \...
-1
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1answer
46 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. ...
9
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1answer
142 views
+50

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 ...
4
votes
2answers
76 views

Showing that $\prod_{k=1} ^{\infty}\left(1- \frac{1}{2k}\right) = 0$ [closed]

How to find this product? $$\prod_{k=1} ^{\infty}\left(1- \frac{1}{2k}\right)$$ I know the answer is $0$, but I want to know how and why.
7
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3answers
108 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 \...
10
votes
2answers
117 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). ...
1
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1answer
58 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 $s$ ...
1
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0answers
34 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}$$\...
1
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3answers
62 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 ...
2
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0answers
16 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 ...
0
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1answer
33 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
38 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
votes
1answer
26 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 ...
0
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3answers
78 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: $$\...
1
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2answers
73 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, ...
0
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0answers
31 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 ...
1
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1answer
26 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 ...
4
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0answers
76 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_{...
1
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1answer
52 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
75 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.
7
votes
8answers
281 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)\...
1
vote
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-...
4
votes
1answer
76 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 $...
1
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3answers
74 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 ...
0
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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 ...
0
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2answers
104 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 ...
1
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2answers
21 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
9 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 $$...
1
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1answer
48 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. ...
1
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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
42 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
59 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
votes
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 $$\sum |1-f(s)|$$ ...
0
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0answers
52 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?
0
votes
1answer
78 views

For each $\beta \in(0,1)$, give a sequence $\{a_j\}\subset(0,1)$ with $\prod (1-\alpha_j) = \beta $

Suppose $\{\alpha_j\} \subset (0,1)$. Give a sequence ${\alpha_j}$ such that $\prod (1-\alpha_j) = \beta $, where $\beta \in (0,1)$ I was able to prove part a. That is the infinite product of $(1-a_j)...
1
vote
1answer
30 views

How to prove the set of bounded sequences is clopen in the uniform metric?

Consider the metric $d(x,y)=\sup\{\overline d(x_i,y_i)|i\in I\}$ on $X=\mathbb R^\omega$ where $\overline d(x,y)=\min\{d(x,y),1\}$ is the standard bounded metric. Consider the topology induced by that ...
0
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0answers
50 views

Manipulating infinite products (Hadamard product for the Riemann Zeta)

In ch. 3 of 'An introduction to the theory of the Riemann Zeta-Function' (S. J. Patterson 1988), it is claimed that $$ s(1-s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s) = Ae^{Bs}\prod_{\rho \in \...
1
vote
0answers
48 views

What topology do we give the set of polynomial functions

Let $X$ be a topology space. $\mathbb C[t]$ be the set of polynomial functions. Define the map $$a: X \rightarrow \mathbb C[t], \quad x \mapsto \sum_{i=1}^n a_i(x) t^i $$ where $a_i:X \...
0
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1answer
41 views

let define the “waved” factorial as $\prod_{i=1}^n (\text{if } \bmod(i,2)=={0\text{ or } 1}\text{ then } [2/i] \text{ else } [i])$

Consider these two products: EvenWavedFactorial = $$\prod_{i=1}^n \text{if } (i\bmod 2 ==0)\text{ then } \left(\frac{2}{i}\right) \text{ else }(i)$$ OddWavedFactorial = $$\prod_{i=1}^n \text{ if } ...
2
votes
1answer
52 views

How does this infinite product work?

${\displaystyle \prod_{n=1}^{\infty} ({ \frac n {n+1}}) ^{(-1)^n}}$ Typed this infinite product into Wolfram Alpha and I got an approximate result of 1.5708. I wonder if anyone studied this infinite ...
2
votes
1answer
136 views

Finding the value of $\prod_{n=0}^\infty a_n$ with $a_0=1/2$ and $a_{n}=1+(a_{n-1}-1)^2$

Putting the value of the first term, we can see that the series goes like $$1/2, 5/4, 17/16,...$$ I am unable to calculate the general term, and so not sum of the series. Please help me to find the ...
0
votes
1answer
37 views

Can $\prod_{n=1}^\infty(1+e^{2\pi i n\tau})$ be expressed in terms of the Euler Phi Function?

I am wondering if $$(-e^{2\pi i \tau},e^{2\pi i \tau})_\infty=\prod_{n=1}^\infty(1+e^{2\pi i n\tau})$$ can be expressed in terms of Euler's Phi Function. Any help is appreciated.
3
votes
0answers
75 views

Evaluating $\sum\limits_{n \ge 0} \frac{1}{x^{2^n}-y^{2^n}}$ where $x, y \in \mathbb R^{+}$ and $x \ne y, x>1.$

I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as $$S = \frac{1}{x-y} \sum_{n ...
3
votes
1answer
88 views

Factorization $\cos\left(\tfrac{\pi z}{4}\right)-\sin\left(\tfrac{\pi z}{4}\right)$

Prove that $$\cos\left(\frac{\pi z}{4}\right)-\sin\left(\frac{\pi z}{4}\right)=\prod_{n=1}^\infty\left(1+\frac{(-1)^nz}{2n-1}\right)$$ My try: $$\cos\left(\frac{\pi z}{4}\right)-\sin\left(\frac{\...
6
votes
0answers
80 views

Proving that the sequence $1, \frac12, \frac{1/2}{3/4}, \cdots$ converges to $\frac{\sqrt 2}2$ [duplicate]

I saw this on a Facebook page: \begin{align*} 1, \qquad \frac12,\qquad\frac{\frac12}{\frac34}, \qquad\frac{\frac{\frac12}{\frac34}}{\frac{\frac56}{\frac78}},\qquad \dots\to \frac{\sqrt{2}}2. \end{...
6
votes
3answers
159 views

Evaluate $\sum_{n=0}^\infty (-1)^n \ln\frac{2+2n}{1+2n}$

I'm trying to evaluate: $$\sum_{n=0}^\infty (-1)^n \ln\frac{2+2n}{1+2n}$$ I'm not too sure where to start. I've tried writing it as a telescoping sum, but that doesn't work. I'm thinking it's ...
1
vote
0answers
64 views

Alternate ways of showing that $\lim_{n \rightarrow \infty} \sum_{n}^{\infty}g_{n}(z_{n}) = \sum_{k}^{\infty} g_{n}(z) $?

In the text "Basic Complex Analysis" Third Edition by Jerrold E. Marsden. I'm inquiring if there's any more alternate approach's to $\text{Proposition (1)}$ and if so I'd like a hint towards a proof ? ...
2
votes
0answers
18 views

Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product \begin{equation*} \prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
0
votes
3answers
145 views

Infinite product $\prod\limits_{n=1}^\infty\cos( 1/n)$

Infinite product $\prod\limits_{n=1}^\infty\cos( 1/n)$ My attempt is to use the complex representation and the Maclaurin Series of $\cos(1/n)$ but I could not find the formula $$\prod_{n=0}^\infty (...