Questions tagged [infinite-product]

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

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A curious infinite product

Let $R$ be the ring $\mathbb{Z[q]}/(q^2)$ whose elements $a+bq$ satisfy $q^2=0.$ Define $g_n\in R$ by $g_1=1$, $g_{2^n}=1-2^{n-1}q$ for $n>0$ and $g_{2n}=0,$ $g_{2n+1}=-q$ else. It seems that ...
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Logarithmic production

Consider the product$$\prod_{n=2}^\infty\left(1+\left(\log_{\sqrt n^{\sqrt n}}t\right)^2\right)=\prod_{n=2}^{\infty}\left(1+\frac{(\log_nt^2)^2}n\right)$$the inner part goes to one for every given $t$,...
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Product representation of the exponential series.

Let ${a_n} = \prod\limits_{j = 1}^n {\gcd (j,n)} .$ Comparing OEIS A067911 and A170911 suggests that there are integers $b_n$ such that the $n-$th partial sum of the product $\prod\limits_{k = 0}^\...
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84 views

Sum $\sum \frac{1}{(4k-3)(4k-2)(4k-1)(4k)}$

I am stuck on this problem for quite a while now, and I don't seem any closer to the solution. So, here it is: $S = 1/4! + 4!/8! + 8!/12! + 12!/16! + ......$ I crossed out the factorials first, and ...
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1answer
29 views

infinite-dimensional inner product space

I've been asked to get an example of an infinite-dimensional inner product space so I wrote this enter image description here as an example is there anything wrong with what I wrote?
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1answer
81 views

Product $\left[\sin(x)\cos\left(\frac{x}2\right)\right]^{1/2}\cdot\left[\sin\left(\frac{x}{2}\right) \cos \left(\frac{x}4\right)\right]^{1/4}\ \cdots$

I came across this question in the following form: Compute the following infinite product $$\left[\sin (x)\cos \left(\frac{x}{2}\right)\right]^{1/2}\cdot \left[\sin \left(\frac{x}{2}\right) \cos \...
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4answers
395 views

Evaluate $\lim_{n\to\infty} \prod_{k=1}^n \frac{2k}{2k-1}\int_{-1}^{\infty} \frac{{\left(\cos{x}\right)}^{2n}}{2^x} \; dx$

Problem 9 in the JHMT 2013 Calculus Test asks to evaluate $$\lim_{n\to\infty} \prod_{k=1}^n \frac{2k}{2k-1}\int_{-1}^{\infty} \frac{{\left(\cos{x}\right)}^{2n}}{2^x} \; dx$$ The answer is $\pi\cdot 2^\...
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Convergence of the product $\prod_{n=1}^{\infty}\left(1+\frac{x^n}{n^p}\right)\cos\frac{x^n}{n^q}$

I have to determine the convergence of the following product (here $p,q\in\Bbb R$). $$ \prod\limits_{n=1}^{\infty}\left(1+\frac{x^n}{n^p}\right)\cos\frac{x^n}{n^q} $$ Denote $a_n=\left(1+\frac{x^n}{n^...
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38 views

If $S_n=\sum _{i=1}^n\frac{1}{a_i}$ and $a_k=\left(\prod _{i=1}^{k-1}a_i\right)+1$ then evaluate $\lim _{n\to \infty }S_n$.

If $$S_n=\sum _{i=1}^n\frac{1}{a_i}$$ and $$a_k=\left(\prod _{i=1}^{k-1}a_i\right)+1$$ then evaluate $$\lim _{n\to \infty }S_n$$ I have tried to simplify a few terms, but it does not seem to cancel ...
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99 views

Does $\prod_{m=1}^\infty \frac{1}{m^2}$ have a closed form?

We know that $$\sum_{m=1}^\infty \frac{1}{m^2} = \frac{\pi^2}{6},$$ but what about the product of the reciprocal of the squares: $$\prod_{m=1}^\infty \frac{1}{m^2}?$$ Do we use a different product ...
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16 views

Conditions to be followed while Infinite Nesting so that the nested expression gives the intended initial expression?

I'd like to start from the problem which lead me to the whole question and then get into the broader question. Given $ \sum _{k=1}^\infty \lfloor \frac n{2^k} \rfloor$ I thought the sum evaluates to ...
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32 views

Metric in which consecutive primes have equivalent gaps?

I thought that maybe the following product could be a good start to look for a metric in which consecutive primes are the same distance apart. $$\frac{1}{\prod_{k=1}^\infty \big(p_k-1\big)^k}$$ ...
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41 views

Why $\frac{\sin(x)}{x} = (1-\frac{x^2}{\pi^2})(1-\frac{x^2}{4\pi^2})…$ [duplicate]

If we allow some $g(x) = \frac{\sin(x)}{x}$ such that $g(0) = 1$ and also some $f(x)$ where: $$f(x) = \prod_{k=1}^{\infty} 1 - \frac{x^2}{\pi^2k^2}$$ Then it is said f and g are equal. But my ...
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36 views

'Multiplicative' integration and 'Riemann products'

Warning! This question has little rigour and is entirely hand wavy crazy blue-sky speculative thinking so I apologise in advance. I was thinking about the gamma function and how it interpolates the ...
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1answer
53 views

Evaluate $\prod_{k=1}^{\infty}\frac{{5^\frac {1}{2}}^k+{3^\frac{1}{2}}^k}{2}$

While doing questions on series and products, I tried hard to solve this infinite product, but no methods has worked so far.Well, this is from Johns Hopkins math tournament Question:- Evaluate $\...
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20 views

Associativity of infinite matrix product.

Many texts reads "It is well known that for infinite matrices multiplication is non-associative". A treatise on this can be found in On the associativity of infinite matrix multiplication. However, ...
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25 views

Weierstrass-type factorization (reconstruction) of a function with branch cuts

Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,...
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1answer
47 views

Unusual Infinite nested radicals

Find the value of this infinitely nested radical in terms of a. Is this even possible? Edit: I believe that I have already solved it, though I would like confirmation that my work is correct. Here ...
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3answers
82 views

Evaluate $\prod_{k=1}^{\infty}\dfrac{(9k^2-1)}{9k^2} \prod_{k=1}^{\infty}\dfrac{(6k-3)^2-4}{(6k-3)^2}$ [closed]

I have been trying to solve a problem, yet when I was doing the work, It came out with $2$ different infinite products. How do I evaluate these infinite products? $$\prod_{k=1}^{\infty}\dfrac{(9k^2-...
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1answer
101 views

proving: $(1+x)(1+x^2)(1+x^3)\ldots = \frac{1}{(1-x)(1-x^3)(1-x^5)\ldots}$

I need to prove: $(1+x)(1+x^2)(1+x^3)\ldots = \frac{1}{(1-x)(1-x^3)(1-x^5)\ldots}$ for $0<x<1$ First i proved: $(1)$ $\prod_{n=0}^{\infty}(1+x^{2^n}) = \frac{1}{1-x}$ for $ |x|<1$ Now ...
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1answer
89 views

Is there any significance to the product $\prod_\limits{n=1}^{\infty} \left(1+\frac{1}{n^x}\right)$?

Is there any significance to this product? $$\prod\limits_{n=1}^{\infty} \left(1+\frac{1}{n^x}\right)$$ Basically taking the Riemann zeta function and trying to make it into a convergent product,...
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1answer
31 views

Proof for the product of an infinite series

Taking for example 2 x 2 x 2 ..., I know that we can represent this as r = 2r, which should result to be r = 0. Now my question arises when attempting to prove whether the (k+1)th element is greater ...
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Volterra product integral identity (proof)

I am reading myself about Product integrals and I find three types of definition. See, for instance, https://en.m.wikipedia.org/wiki/Product_integral I am interested on the type I product integral. ...
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158 views

On the formula, $\pi = \frac 5\varphi\cdot\frac 2{\sqrt{2+\sqrt{2+\varphi}}}\cdot\frac 2{\sqrt{2+\sqrt{2+\sqrt{2+\varphi}}}}\cdots$

I found a formula on google images when I was looking at some formulas for $\pi$ just for the fun of it, and I came across one that really startled me, and was quite reminiscent of Viète's product. ...
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1answer
50 views

Convergence of an infinite product (2)

Consider this infinite product: $$\prod _{n=1}^\infty \Bigl( 1+ \frac z n\Bigl) e^{-\frac z n}\ .$$ I must show that it converges to an entire function $f$, and determine the zeros of $f$. It's ...
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55 views

Evaluate $\lim_{n \to \infty}\prod_{k=0}^{n} \left(1+\frac{2}{45^{2^k}+45^{-2^k}}\right)$

Evaluate $$P=\lim_{n \to \infty}\prod_{k=0}^{n} \left(1+\frac{2}{45^{2^k}+45^{-2^k}}\right)$$ My try: Let $a_k=45^{2^k}$ Then we have $45^{-2^k}=\frac{1}{a_k}$ So We get: $$1+\frac{2}{a_k+\frac{1}...
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Is infinite product with this characteristic convergent?

Given the infinite product of: $$\prod_{n=k}^{\infty} a_n$$ If I could prove that $\lim_{n\to\infty} a_n=1$, does it follow that the aforementioned infinite product converges?
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2answers
21 views

Circumscription to infinity: Is this infinite product convergent?

A circle with radius $r_1=1$ is inscribed in an equilateral triangle. The triangle itself is inscribed in a larger circle and this larger circle is inscribed in a square. If we continue this process ...
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2answers
59 views

Closed form for the infinite product $S_n=\Pi_{i=1}^{\infty} (1+\frac{1}{2^i})$

Consider the infinite product $S_n=\Pi_{i=1}^{n} (1+\frac{1}{2^i})$. The original problem was to prove that $\Pi_{i=1}^{n} (1+\frac{1}{2^i}) < \frac{5}{2} \ \forall \ n \in \mathbb{Z^+}$. This can ...
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3answers
96 views

Computing $\lim_{n\to\infty} \prod_{k=1}^n(1-\frac{x^2k^{2\alpha}}{n^{2 \alpha+1}})$

Let $\alpha>0,x \in \mathbb{R}$ I am having a problem in computing the following limit: $$\lim_{n \to \infty} \prod_{k=1}^n\bigg(1-\frac{x^2k^{2a}}{n^{2a+1}}\bigg).$$ In fact: the problem was ...
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1answer
83 views

how to calculate $ \prod_{n=1}^\infty\left(1\pm q^{2n-1}\right), \prod_{n=1}^\infty\left(1\pm q^{2n}\right)$

It is well known that we have the following pentagon number theorem by Euler: $\prod_{n=1}^\infty\left(1-q^{n}\right)=\sum_{-\infty}^{\infty}(-1)^nq^{\large \frac{3n^2-n}2}$. However, how to ...
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47 views

What value does $\prod_{n=0}^\infty 1+{1\over x^n}$ converge to?

I would like to find out the exact value (if it can be written down and generalized) for the formula in the title for any x. I have got aproximate values, but I would apreciate an exact value of a ...
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1answer
38 views

An infinite product obtained by taking every third term from the Wallis product. Is it new?

Numerical calculation suggests that $\prod_{n=0}^\infty \frac{(6n+2)^2}{(6n+1)(6n+3)} = \frac{3^{1/2}}{2^{1/3}}$. Has this been proven?
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1answer
86 views

Infinite product for sin z

Assuming that $$sin z = {z}\prod_{r=1}^{\inf}(1-({z^2}/{r^2}{\pi^2})),$$ show that if $$m\rightarrow \inf$$ and $$ n\rightarrow \inf $$ in such a way that lim (m/n) = k where k is finite,then $$ lim\...
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1answer
50 views

When is the Infinite product of continuous functions a continuous function? Assume that the product is convergent.

Is there any theorem about the continuity of an infinite product of continuous real valued functions on compact Housdorff spaces, if the product is convergent? I mean, for each natural number $n$, let ...
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37 views

Not proving the irrationality of $e^e$

How would one go about proving that there does not exist such a product that satisfies $$ e^k=\prod_{n=1}^{k}\frac{a_n}{b_n} $$ for $k$, $a_n$, $b_n\in\mathbb{N}$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=...
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1answer
121 views

Prove $\prod_{n \in \Bbb{N} } \left(\frac{1-q^n}{1+q^n}\right)^{(-1)^n} = \sum_{n \in \Bbb{Z}} q^{n^2}$

Let $q$ be a complex number with $|q| < 1$, prove that $$ \prod_{n \in \Bbb{N} } \left(\dfrac{1-q^n}{1+q^n}\right)^{(-1)^n} = \sum_{n \in \Bbb{Z}} q^{n^2} $$ Not sure if this helps but the LHS can ...
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59 views

Calculating $\prod\limits_{k=0}^\infty(1+{\rm e}^{-(2k+1)\pi})$

Someone computed it by Mathematica and got that $$\prod_{k=0}^\infty(1+{\rm e}^{-(2k+1)\pi})=2^{\frac14}{\rm e}^{-\frac\pi{24}}.$$ However, I don’t know how to prove it. I would appreciate it if ...
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43 views

Demonstrate this inequality using weak induction [closed]

Demonstrate this inequality using weak induction on $n$: For all $n\in\Bbb N^{\ge 1}$, $$\prod_{i=1}^n\frac{2i-1}{2i}\le\frac1{\sqrt{3n+1}}$$
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1answer
35 views

Limit of the infinite product [closed]

How can I prove this result, $\lim\limits_{N\rightarrow\infty} \ \prod\limits_{i = 0}^{N} \left(1-\frac{a_i}{N}\right) = e^{\lim_{N\rightarrow\infty}-\frac{1}{N}\sum_{i=0}^{N} a_i}$ for $a_i \in O(...
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1answer
60 views

Show convergence of infinite product

Let $w \in \mathbb{C}$ be a complex number with $0 <|w|<1$. Show that the infinite product $$\theta(z) := \prod_{n=1}^\infty (1+w^{2n-1}e^z)(1+w^{2n-1}e^{-z})$$ converges locally uniformly and ...
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1answer
17 views

Borel sigma algebra on uncountable product and product sigma-algebra

Let $\Omega_{i}$ be metric spaces for $i \in I$ (uncountable). Consider two $\sigma$-algebras on the product space $\Omega=\prod_{i \in I}\Omega_{i}$: The Borel $\sigma$-algebra on $\Omega$ which is ...
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1answer
38 views

If $F(n)$ is the unique number of arranging $n \ge 1$ unique items, prove inductively that $F(n) = n!$

This question doesn't seem so hard. Then again it does. I'm struggling to move past the induction hypothesis. The base case is obviously $1!$. Assuming that $F(n) = n!$ , I take the $(n+1)!$ ...
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1answer
63 views

When the product $|\prod_{k=1}^n(1-z^k)|$ tends to infinity?

Let $z\in\mathbb{C}$. I would like to know when the product $$|\prod_{k=1}^n(1-z^k)|$$ tends to infinity? My attempt : We have: $$|\prod_{k=1}^n(1-z^k)|= \prod_{k=1}^n |1-z^k|$$ Since we have: ...
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3answers
84 views

Find value of $\prod_{k=0}^{2^{1999}}\left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right)$

Find value of $$S=\prod_{k=0}^{2^{1999}}\left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right)$$ We have for $k=0$ the value as $-3$ and now for $k \ne 0$ $$S_1=\prod_{k=1}^{2^{1999}}\left(\frac{\...
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1answer
47 views

How to use Euler formula to prove the following conclusion? [duplicate]

I thought about it for a long time $$\prod_{k=1}^{n}\sin(\frac{k\pi}{2n})=\frac{\sqrt{n}}{2^{n-1}}$$
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1answer
95 views

Find the limit of an infinite product related to the alternating harmonic convergence.

$$ \lim_{x\to\infty}\prod_{n=1}^x \left(1+\frac1{xn(2n-1)}\right)^x=4 $$ I know this equals 4 because I derived this equation from the alternating harmonic series: $1-\frac1{2} +\frac1{3} -\frac1{4} +...
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1answer
37 views

Is there a closed form of $\prod_{p \in \mathbb{P}} p^{p^{-k}}$ ,$k >1$?

I have did a search in web to get the closed form of this product $\prod_{p \in \mathbb{P}} p^{p^{-k}}$, $k >1$ and $k$ is a real number such that I have selected topics related to Euler product ...
4
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1answer
72 views

Show that: $\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$

How to show that: $$P=\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$$ Where $\phi=\frac{1+\sqrt{5}}{2}$ $$P=\frac{\sqrt{5}F_{4}+1}{\sqrt{5}F_{4}-1}\cdot \frac{\sqrt{5}F_{6}+1}{...
3
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1answer
58 views

Relation between lemniscate constant and hyperbolic tangent.

Recently I came across this identity: $$ \prod _{n=1}^{{\infty }}\:\tanh \:\left(\frac{\pi n}{2}\right)=\frac{\Gamma \left(\frac{1}{4}\right)}{\sqrt[4]{8\pi ^3}} $$ The question is: how to prove this ...

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