# 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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### 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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### 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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### Demonstrate this inequality using weak induction [closed]

Demonstrate this inequality using weak induction on $n$: For all $n\in\Bbb N^{\ge 1}$, $$\prod_{i=1}^n\frac{2i-1}{2i}\le\frac1{\sqrt{3n+1}}$$
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 $a_i \in O(... 1answer 60 views ### Show convergence of infinite product Let$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 ... 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 ... 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$(n+1)!$... 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: ... 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\$ $$S_1=\prod_{k=1}^{2^{1999}}\left(\frac{\... 1answer 47 views ### How to use Euler formula to prove the following conclusion？ [duplicate] I thought about it for a long time$$\prod_{k=1}^{n}\sin(\frac{k\pi}{2n})=\frac{\sqrt{n}}{2^{n-1}}$$1answer 95 views ### Find the limit of an infinite product related to the alternating harmonic convergence.$$ \lim_{x\to\infty}\prod_{n=1}^x \left(1+\frac1{xn(2n-1)}\right)^x=4 $$I know this equals 4 because I derived this equation from the alternating harmonic series: 1-\frac1{2} +\frac1{3} -\frac1{4} +... 1answer 37 views ### Is there a closed form of \prod_{p \in \mathbb{P}} p^{p^{-k}} ,k >1? I have did a search in web to get the closed form of this product \prod_{p \in \mathbb{P}} p^{p^{-k}}, k >1 and k is a real number such that I have selected topics related to Euler product ... 1answer 72 views ### Show that: \prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi How to show that:$$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}{...
Recently I came across this identity: $$\prod _{n=1}^{{\infty }}\:\tanh \:\left(\frac{\pi n}{2}\right)=\frac{\Gamma \left(\frac{1}{4}\right)}{\sqrt{8\pi ^3}}$$ The question is: how to prove this ...