Questions tagged [infinite-product]

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

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How to compute $\displaystyle\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}}$?

In my text book it is stated (without any explanation) that $$ \prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}} = e^{e^x - 1} $$ and I can't really think of how one can show this.
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Infinite product converging arbitrarily slowly to limit

There evidently exist infinite series that converge to their limits with arbitrary speed. For example, for each $\alpha>0$, the series $$ \sum_{j=1}^n j^{-\alpha-1} \to \sum_{j=1}^\infty j^{-\alpha-...
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Factorization of modular forms using zeros and poles

In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ...
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Proving that this infinite product is convergent

This question was asked in my assignment in number theory and I could not prove it. Question : Define the multiplicative function w(n) such that $w(p^k) =0$ for $k\geq 2$ and w(p)= { $\frac{p} { f(p)...
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For what infinite series can a closed form be obtained by means of the $\text{Sum} = \text{Product} $ method?

Euler solved the Basel problem by equating the Taylor series and the infinite product representation of $\sin(x)/x$: $$\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n+1)!} = \prod_{k=1}^{\infty}\bigg{(} ...
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If $\alpha$ is the nth root of unity , then the value of [closed]

Given , $\alpha$ is the nth root of unity . Then , the value of $$(11-\alpha)(11-\alpha^2)(11-\alpha^3)........(11-\alpha^{n-1})$$ is equal to ... I tried to use the property : That sum of nth roots ...
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Summing $\sum_{k=1}^{\infty} \frac{(-2)^k + 1}{3^k}$

I’m trying to calculate the sum of this by breaking it down into two geometric series: $$\sum_{k=1}^{\infty} \frac{(-1)^k2^k + 1}{(3^k)} =\sum_{k=1}^\infty\left(-\frac{2}{3}\right)^{k} +\sum_{k=1}^\...
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How to calculate $\prod _{n=1}^{\infty}\frac{2^n-1}{2^n}$?

Suppose we have a kind of lottery as follow: $1.$ You have a $\frac{1}{2}$ possibility of getting a prize on the first try. $2.$ You have a $\frac{1}{4}$ possibility of getting a prize on the second ...
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Find a formula for calculate $\prod_{n=0}^N(a^n + b)$

I have an expression, in which I am looking for a formula to calculate $$ \prod_{n=0}^N (a^n +b)$$ as a summation. Is there a general name for this type of formula? Or a relevant approach to find ...
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Countable product of complete spaces is complete

Consider Fréchet spaces $\{E_n\}_{n\in\mathbb{N}}$, and let $E = \prod_{n=1}^{\infty} E_n$. Show that $E$ is a Fréchet space. I'm stuck with proving completeness. I have already verified that $E$ is ...
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Factorial limit convergence on e

I'm working with some particular infinite products. Each infinite product applies to a range of real numbers, in much the same way as a Riemann sum applies to a range of real numbers. The specifics of ...
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Is the sum of norms/metrics a norm/metric on the product space?

Let be $\mathfrak X:=\Big\{\big(X_i,\nu_i\big):i\in I\Big\}$ a FINITE collection of normed spaces. So I would like to know if the position $$ \nu(x):=\sum_{i\in I}\nu_i(x_i) $$ for all $x\in\prod_{i\...
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Why is $\lim\limits_{n\to\infty} \prod\limits_{k=1}^n \left(1 + \frac{k+1}{n^2}\right) = \sqrt e$?

Mathematica returns these somewhat striking (to me, at any rate) infinite product identities: $$\lim_{n\to\infty} \prod_{k=1}^n \left(1 + \frac{k+1}{n^2}\right) = \sqrt e$$ and $$\lim_{n\to\infty} \...
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An infinite product

So there is this product $\frac{2}{1} \times \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5}...$ I got its expression of a general term but converting it into a sum by a logarithm doesn't help ...
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Closed form of $f(z) = \prod_{n\in\mathbb Z} \frac{z-n-\bar\alpha}{z-n-\alpha} \times \frac{-z-n-\alpha}{-z-n-\bar\alpha}$

For $\alpha\in\mathbb C$ with $\mathrm{Im}(\alpha)>0$, consider the infinite product $$f(z) = \prod_{n\in\mathbb Z} \frac{z-n-\bar\alpha}{z-n-\alpha} \times \frac{-z-n-\alpha}{-z-n-\bar\alpha}.$$ ...
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Is there any known identities for $\prod_{n\in\mathbb Z} \frac{z^2 +(n+\alpha)^2}{(n+\alpha)^2}$?

Let $\alpha\in\mathbb R -\mathbb Z$, and let $z\in\mathbb C$. Is there any closed form expression for either one of the two infinite products: $$\prod_{n\in\mathbb Z} \frac{z^2 +(n+\alpha)^2}{(n+\...
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Limit of infinite product $\lim_{x\to\infty} \prod_{n\in\mathbb Z} \frac{(x-1)^2+n^2}{(x+1)^2+n^2}$

What is the limit $$\lim_{x\to\infty} \prod_{n\in\mathbb Z} \frac{(x-1)^2+n^2}{(x+1)^2+n^2} \:?$$ The infinite product is well-defined pointwise since $\frac{(x-1)^2+n^2}{(x+1)^2+n^2}-1 = O(1/n^2)$ ...
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2 votes
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How can I argue that an infinite sum in this form diverges?

I came across this infinite sum (or infinte product, or series?): $$\sum_{n=1}^{\infty }\frac{3n^2+7}{9n^3+4n-1}$$ which diverges to infinity. But how can I "prove" that? If the equation ...
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Convergence of infinite product to exponential

Let $x_n\to x$ be a convergent sequence of real numbers with limit $x\in\mathbb R$. Then $$\lim_{n\to\infty}\prod_{k=1}^n \left(1+\frac{x_k}{n}\right)=e^x.$$ Is this true? (Note: The formula has $n$ ...
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Proof of uniform convergence of an infinite product

I am reading the book Complex Analysis: An Invitation (2nd Edition), page 163-164. There is a certain step in the proof, which I can not fill the details. First, I mention a relevant proposition and a ...
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4 votes
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Properties of the convergence of the infinite products

Let $(a_n)_{n\geq 1}$ be a sequence of complex numbers. The infinite product $\prod_{k=1}^\infty a_k$ is said to converge if for all $\epsilon>0$ there is $N\in\mathbb{N}$ such that $|a_{n+1}\...
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1 answer
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calculate limitation of product of expression [closed]

I'm pretty sure the limitation value exists for the following expression. I have tried to factorize each item but to no avail. Any suggestions? $(1-1/2)(1-1/4)(1-1/8)(1-1/16)$...
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Solution verification of $\prod_{n=1}^{\infty} a_n$ and $\prod_{n=1}^{\infty} b_n$ convergent, what can be said about $\prod_{n=1}^{\infty} a_n^2$

Assume that the products $\prod_{n=1}^{\infty} a_n$ and $\prod_{n=1}^{\infty} b_n$ with positive factors both converge. Study the convergence of convergence of: (a) $\prod_{n=1}^{\infty} a_n^2$; (b) $\...
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What can we say about the infinite product $\prod_{n = 1}^\infty (1 - x/n^p)$?

Consider the infinite product $$P(x,p) = \prod_{n = 1}^\infty (1 - \dfrac{x}{n^p})$$ where $x \geq 0$ and $p \geq 0$. Does it have a closed form? Are there general values for $x, p$ such that this ...
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1 vote
1 answer
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Proving that $\prod_{n=0}^\infty (1 + x^{2^{n}}) = \frac{1}{1-x}$

I would like to verify the truth of my proof that $$\large\prod_{n=0}^\infty (1 + x^{2^{n}}) = \frac{1}{1-x}.$$ Proof. The proof is rather straightforward, just using some observations that two ...
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1 answer
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Convergence of $\prod_{n=1}^{\infty} (1 - z^n)$ when $|z| = 1$

Clearly, $\prod_{n=1}^{\infty} (1 - z^n)$ converges to $0$ when $1 - z^{n_0} = 0$ for some $n_0$. Suppose $z$ is a complex number with $|z| = 1$ and $1 - z^{n} \ne 0$ for all $n$. I think $\prod_{n=1}^...
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Two competing definitions of the direct sum of vector spaces

There seem to be two competing definitions of the direct sum of vector spaces. The first one characterises it as the same as the Cartesian product for a finite number of vector spaces, and for an ...
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1 vote
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Sum of Reciprocal of Repeated Products [closed]

For what real values of $c\in\mathbb{R}$ does the following series converge and diverge? $$ \sum_{k=1}^\infty\frac{1}{\prod\limits_{\substack{1\leq j\leq k \\ j\neq -c}}\left(1+\frac{c}{j}\right)} $$
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Is the following infinite product of fractions of linear factors equal to an exponential function or not?

Is the following infinite product: $$ \prod_{\substack{(a,b) \in \mathbb{Z}^2 \\ a > b}} \frac{x+b}{x+a} $$ defined? If so, does it simplify to an exponential function of $x$? The subscript is ...
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2 votes
2 answers
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How to compute $\lim_{n \to \infty} \frac{1}{n} \left[(n^2 +1^2)(n^2+2^2)^2 \cdots (n^2 + n^2)^n \right]^{\frac{1}{n^2}}$

Trying to prove $\lim_{n \to \infty} \frac{1}{n} \left[(n^2 +1^2)(n^2+2^2)^2 \cdots (n^2 + n^2)^n \right]^{\frac{1}{n^2}} = 2e^{-1/2}$ I tried using ratio-root criteria with $a_n= \frac{1}{n^n} \...
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2 votes
0 answers
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Is the fact that $\int_{0}^{\infty} \cos(2x)\prod_{n=1}^{\infty} \cos\big(\frac{x}{n}\big) dx\approx \frac{\pi}{8}$ a pure numerical coincidence?

I was reading about mathematical coincidences recently, and I came across the borderline unbelievable fact that $$\int_{0}^{\infty} \cos(2x)\prod_{n=1}^{\infty} \cos\Big(\frac{x}{n}\Big) dx\approx \...
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1 answer
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Suppose $A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)$. If $A$ and $f(k)$ are known, how to find $B$ and $g(n)$?

My question like "some-to-product" or vice versa. See the following example (for reference, see here and here); $$\pi=\sum_{k=0}^{\infty}\frac{4(-1)^k}{2k+1}=2\prod_{n=0}^{\infty}\frac{4n^2+...
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Infinite product of $(1/2)\cdot(3/4)\cdot(7/8)\cdot(15/16)\cdot(31/32)\cdot\;\cdots\,$? [duplicate]

This problem has been bothering me for a while. I know that $$(1/2)\cdot(3/4)\cdot(7/8)\cdot(15/16)\cdot(31/32)\cdot\;\cdots$$ is convergent to something other than zero (around $0.2887$), but I do ...
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2 votes
1 answer
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The maps $f_n: \Bbb R^n \to \Bbb R^{\Bbb N}$ induce final topology $\tau$ on $\Bbb R^\infty$. Compare it with the subspace top from $\Bbb R^{\Bbb N} $

Let $\Bbb R^\infty$ be the set of sequences in $\Bbb R^{\Bbb N}$ for which $x_j \ne 0$ for finitely many $j$. The maps $f_n:\Bbb R^n \to \Bbb R^{\Bbb N}$ where $f_n(x)=(x_1,x_2, \dots x_n,0,0 \dots)$ ...
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2 votes
2 answers
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Solving an infinite alternating product

Idea is to alternately multiply infinite many values from $1$ to $x$, with const distance $h$. Reweriting it to math: $$ h = \frac{x-1}{n} $$ $$ f(x) = \lim_{n \to \infty} \left( \prod_{m=0}^{n} (x - ...
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Prove that $\prod_{i\in I}X_i\cap\prod_{i\in I}Y_i=\prod_{i\in I}(X_i\cap Y_i)$

So if $X$ and $Y$ are two set then any binary relation between $X$ and $Y$ is a subset of the power set $\mathcal P\big(\mathcal P(X\cup Y)\big)$ of $\mathcal P(X\cup Y)$ so that we can define the set ...
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3 votes
1 answer
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Evaluate $\lim_{n \to \infty} \prod_{k=n+1}^{2n} k^{1/k}$

Problem: evaluate $\lim_{n \to \infty} \prod_{k=n+1}^{2n} k^{1/k}$. My work: since $n+1 \le k \le 2n$, it is $\frac{1}{2n} \le \frac{1}{k} \le \frac{1}{n+1}$. Since $k \ge 1$, the exponential with ...
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How to compute this infinite product

I have this product $$ C_t(n)=\prod_{0 < q <t \land q\neq n} \frac{1}{e^{\frac{i\pi q}{t}}-e^{\frac{i\pi n}{t}}} $$ Where $t,n\in\mathbb{N}$. What value will $C_\infty(n)$ tend to? My computer ...
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4 votes
1 answer
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Proof of a closed-form of $\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}$

I'm looking for a proof of the following equality: $$\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}=\sqrt{\frac{\Gamma\left(\frac{1}{3}\right)}{2\pi}}\frac{3^{13/24} \exp \left[...
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15 votes
2 answers
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Divide a ball of volume $\frac{e^2}{6}n$ into $n$ slices of equal height. What is the product of the volumes of the slices as $n\rightarrow\infty$?

Divide a ball of volume $\frac{e^2}{6}n$ into $n$ slices of equal height, as shown below with example $n=8$. What is the limit of the product of the volumes of the slices as $n\rightarrow\infty$? (If ...
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1 vote
2 answers
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Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ...
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16 votes
1 answer
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Product of areas in a circle

A circle (or disk) of area $2n$ is divided into $n$ regions, as shown below with example $n=8$. The points are evenly spaced around the circle. (If the image doesn't load for you, just imagine $n$ ...
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Finding the value of the infinite product $\prod\limits_{n=1}^{\infty} \biggl(1-\frac{1}{2^{n}}\biggr)$ [duplicate]

The math teacher of a friend of mine gave him this question on a test. He then asked me if I could solve it, but I had no idea how to begin. It goes like this. Find the value of the following infinite ...
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Infinite product over primes' convergence with rearrangement

If we know that $W_m(k)$ is multiplicative over $k$ (if $\gcd(k_1,k_2)=1$ then $W_m(k_1k_2)=W_m(k_1)W_m(k_2)$), and $W_m(k)=0$ if $p^2|k$, and that $|W_m(k)|\le c\cdot \frac{m}{k^{3/2}}$. So that the ...
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0 votes
1 answer
81 views

What is the corresponding infinite series for this infinite infinite product?

What is the infinite series corresponding to the infinite product below? $$f(\alpha,x)=\prod_{n=1}^\infty \left(1+\frac{x}{\alpha^n}\right)$$ Edit: Martin R told me to speak about what I have gotten ...
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1 vote
0 answers
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Derive Xi Function's Infinite Product

The Infinite Product of the Xi Function is $\xi(s)={1\over 2}\prod_{\rho}^\ (1- {s\over \rho})$. But I don't know how to derive it, I know how to derive the Product formula for the zeta function but I ...
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0 answers
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Product representation of some exponential functions?

I was looking at some old questions of mine and stumbled upon this quesiton, which I could not solve: Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The ...
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4 votes
1 answer
219 views

Generalizing Euler's infinite product of cosines

The following formula is attributed to Euler: $$\frac{\sin(2 k)}{2 k} = \prod_{n=0}^\infty \cos(k \frac{1}{2^n})$$ This can be shown through $m$ applications of $\sin(x) = 2 \sin(x/2) \cos(x/2)$ to ...
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0 votes
0 answers
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How caw we write the product space and the corresponding $\sigma-$algebra of this space?

With respect to my new post in here, I have updated the question. I want to define the space of a stochastic game and it confuses me a lot, so I want someone to check please the space $\Theta$ that I ...
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2 answers
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closed form for product of series

Is there a formula expressing the coefficients of the product $$\prod_{k=1}^n \left(\sum_{j=0}^\infty (j+1)x^{jk}\right),$$ where $n\ge 1$, in terms of sums of products of integer partitions?
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