Intereting Posts

Prove that $h'(t)=\int_{a}^{b}\frac{\partial\phi}{\partial t}ds$.
Point on circumference a given distance from another point
Co-countable Topology On Uncountable Set
Is there exist a homemoorphism between either pair of $(0,1),(0,1],$
Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite
If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent.
Formal power series ring, norm.
How does one solve this recurrence relation?
Continuous coloring of a Mandelbrot fractal
Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?
In this isosceles right angled triangle, prove that $\angle DAE = 45^{\circ}$
Ring of integers is a PID but not a Euclidean domain
How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/x))\,dx$?
Ricci SCALAR curvature
How to prove $D^n/S^{n-1}\cong S^n$?

How to prove that $$\displaystyle\prod_{n=0}^\infty \frac{(4n+2)^2}{(4n+1)(4n+3)}=\sqrt{2}$$

Thanks in advances.

- Infinite Product is converges
- Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
- Prove $\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$
- Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$
- Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$.
- Evaluate an infinite product in a closed form

- Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$
- Unnecessary step, normally convergent product (p10, Remmert)
- $\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation
- What's the limit of $\prod_1^\infty \left(1-\frac{1}{2^n}\right)=(1-1/2)(1-1/4)(1-1/8)…$?
- Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $
- Infinite Product Representation of $\sin x$
- Question on $\Pi_{n=1}^\infty\left(1-\frac{x^a}{\pi^an^a}\right)$ and the Riemann Zeta function
- How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?
- Evaluating $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$
- Finding the limit of roots products $(\sqrt{2}-\sqrt{2})(\sqrt{2}-\sqrt{2})(\sqrt{2}-\sqrt{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt{2})$

Rewrite a partial product as

$$\prod_{n=0}^N \frac{(4 n+2)^3 (4 n+4)}{(4 n+1)(4 n+2)(4 n+3) (4 n+4)} = \frac{2^{3 N+3} (2 N+1)!!^3 4^{N+1} (N+1)!}{(4 N+4)!} $$

Use the fact that

$$(2 N+1)!! = \frac{(2 N+1)!}{2^N N!} $$

and

$$M! \approx M^M e^{-M} \sqrt{2 \pi M} \quad (M \to \infty)$$

The rest is careful bookkeeping, and making sure you use the fact that

$$\lim_{M \to \infty} \left ( 1+\frac1{M} \right )^M = e$$

the sought-after result follows.

From the Weierstrass product for the cosine function we have:

$$ \cos z = \prod_{n\geq 0}\left(1-\frac{4z^2}{(2n+1)^2\pi^2}\right) $$

and by taking $z=\frac{\pi}{4}$ it follows that:

$$\prod_{n=0}^{+\infty}\left(1-\frac{1}{(4n+2)^2}\right)=\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$$

whose LHS is just the reciprocal of our product.

**Hint:** Rewrite the infinite product expression in terms of the $\Gamma$ function, and then apply Euler’s famous reflection formula.

- Density of orthogonal matrices with rational coefficients
- Follow-up Question: Proof of Irrationality of $\sqrt{3}$
- Can a number have infinitely many digits before the decimal point?
- number of subsets of even and odd
- Honest and Deceitful Professors Problem
- Group identities and inverses
- Why is this polynomial irreducible?
- Question about Fredholm operator
- Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
- Cartesian Product of Two Countable Sets is Countable
- Meaning of different Orders of Derivative
- Geometric basis for the real numbers
- Expected rank of a random binary matrix?
- Maximal ideals and the projective Nullstellensatz
- We Quotient an algebraic structure to generate equivalence classes?