Newbetuts

Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

Another evaluating limit question: $\lim\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$

Limit as $n\to+\infty$ of $\prod_{k=1}^{n} \frac{2k}{2k+1}$

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

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

How to prove $\prod_{k=1}^{\infty}{\frac{p_k^2+1}{p_k^2-1}}=\frac{5}{2}$?

Product of areas in a circle

Find $\lim\limits_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)$ [duplicate]

Proof of a closed-form of $\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}$

How to prove $ \prod_{n=1}^{\infty} \left(1+\frac{2}{n}\right)^{(-1)^{n+1}n} \,= \frac{\pi}{2e}$

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

Proving $\prod_{n=1}^{\infty}\left(1+\frac{1}{n^{3}}\right)=\frac{1}{\pi}\cosh\frac{\pi\sqrt{3}}{2}$

Convergence/Divergence of infinite product

What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$

sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

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 $n\rightarrow\infty$?

New posts in infinite-product

Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

Limit of product with prime numbers

Anybody know a proof of $\prod_{n=1}^\infty\cos(x/2^n)=\sin x/x$. [duplicate]

Astonishing: the sum of two infinite products of nested radicals equal to $ \pi $.

Another evaluating limit question: $\lim\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$

Limit as $n\to+\infty$ of $\prod_{k=1}^{n} \frac{2k}{2k+1}$

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

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

How to prove $\prod_{k=1}^{\infty}{\frac{p_k^2+1}{p_k^2-1}}=\frac{5}{2}$?

Product of areas in a circle

Find $\lim\limits_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)$ [duplicate]

Proof of a closed-form of $\prod_{n=1}^{\infty} \frac{1}{e} \left(1+\frac{1}{3n}\right)^{3n+1/2}$

How to prove $ \prod_{n=1}^{\infty} \left(1+\frac{2}{n}\right)^{(-1)^{n+1}n} \,= \frac{\pi}{2e}$

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

Proving $\prod_{n=1}^{\infty}\left(1+\frac{1}{n^{3}}\right)=\frac{1}{\pi}\cosh\frac{\pi\sqrt{3}}{2}$

Convergence/Divergence of infinite product

What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$

sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

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 $n\rightarrow\infty$?