Questions tagged [infinite-product]

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

In mathematics, for a sequence of complex numbers $a_1, a_2, a_3, ...$ the infinite product $$ \prod_{n=1}^{\infty} a_n = a_1 \times a_2 \times a_3 \cdots $$ is defined to be the limit of the partial products $a_1a_2... a_n$ as $n$ increases without bound. The product is said to converge when the limit exists and is a non-zero number. If the limit is zero or does not approach a complex number the product is said to diverge: this is to preserve the analogy with positive terms where if both sides converge we have

$$ \prod_{n=1}^{\infty} a_n = \exp\left(\sum_{n=1}^{\infty} \log(a_n)\right) $$

Euler products are a special case where the product is indexed over prime numbers; see for more information.

Not to be confused with , , etc.

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$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation

Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
user2378
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Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root,…
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Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$. I couldn't do it and my friend justified…
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Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$

How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$
Michael Li
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Find the value of $\sqrt{10\sqrt{10\sqrt{10...}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it simplifies to $x=10\sqrt{x}$ which has solutions…
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Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

Is there a closed form for the following infinite product? $$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
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Hahn-Banach From Systems of Linear Equations

In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$: \begin{align*} \begin{array}{ccccccccc} 1 & = & x_1 & +…
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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 $$\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$$ which looks like unbelievable. (I forgot…
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How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?

Does $$p=\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$$ have any closed form in terms of known mathematical constants? The computer says $$p=3.682154\dots$$ but I don't even know how do devise the converging upper and lower bounds to obtain…
Nikolaj-K
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Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
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Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln…
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Manual calculation doesn't match Wolfram Alpha. Why?

Say I want to evaluate this sum: $$\sum_{x=2}^\infty \ln(x^3+1)-\ln(x^3-1)$$ We can rewrite the sum as $$\ln\left(\prod_{x=2}^\infty \frac{x^3+1}{x^3-1}\right)$$ We can split the product into two products: $$\ln\left(\prod_{x=2}^\infty…
D.R.
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Prove that $\prod_{n=2}^\infty \frac{1}{e^2}\left(\frac{n+1}{n-1}\right)^n=\frac{e^3}{4\pi}$

I friend of mine sent me this problem a while ago, but I still can't figure it out. (He can't figure it out either.) I figured here would be a good place to ask for help. Prove: $$\prod_{n=2}^\infty…
Spot
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Convergence of sequence: $ \sqrt{2} \sqrt{2 - \sqrt{2}} \sqrt{2 - \sqrt{2 - \sqrt{2}}} \sqrt{2 - \sqrt{2 - \sqrt{2-\sqrt{2}}}} \cdots $ =?

In other words, if we define a sequence $$ \displaystyle a_{n+1} = \sqrt{2-a_n}, \,\,\,a_0 = 0 .$$ Then, we need to find $$ \displaystyle \prod_{n=1}^{\infty}{a_n}. $$ Well, from here I don't seem to follow. I can understand that there would be…
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Result of the product $0.9 \times 0.99 \times 0.999 \times ...$

My question has two parts: How can I nicely define the infinite sequence $0.9,\ 0.99,\ 0.999,\ \dots$? One option would be the recursive definition below; is there a nicer way to do this? Maybe put it in a form that makes the second question easier…
Paul Manta
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