Intereting Posts

order of operations division
Differentiation with respect to a matrix (residual sum of squares)?
For $x, y \in \mathbb{R}$, define $x \sim y $ if $x-y \in \mathbb{Q}$. Is $\mathbb{R}/\!\!\sim$ Hausdorff?
The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series
Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$.
Upper bound on cardinality of a field
How do you formally prove that rotation is a linear transformation?
Rigorous nature of combinatorics
Irrational numbers to the power of other irrational numbers: A beautiful proof question
Divergent Series Intuition
Is $\mathbb R^2$ a field?
Computing $\int_C \frac{8e^{iz}}{ z^2+i} dz,$ where $C = \{z \in \mathbb{C} : |z| = 3\}$.
How to find positive integer solution of bilinear transformation?
Topological K-theory references
Interchange supremum and expectation

What will be the value of the following ** Infinite Product** :

$$\displaystyle \prod_{k=0}^\infty \left(1+\dfrac{1}{2^{2^k}}\right)$$

It would be nice if anyone could spare the time and boil down to the absolute basics and tell how they reached the solution.

- This one weird infinite product can define exponentials in terms of itself. What does it do for other constants?
- What is the infinite product of (primes^2+1)/(primes^2-1)?
- How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?
- Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$
- Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?
- Evaluating $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$

- Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$
- Closed form for $\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{n\pi}-1}-16^k\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{4n\pi}-1}$
- Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$
- Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number
- Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?
- Finding the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$.
- Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts?
- Sequence is periodic $x_{n+2}=|x_{n+1}-x_{n-1}|$
- how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?
- Any closed subset of $\mathbb C$ is the set of limit points of some sequence

**hint:** $1+\dfrac{1}{2^n} = \dfrac{1-\dfrac{1}{2^{2n}}}{1-\dfrac{1}{2^n}}, n = 2^k$, and realize “product telescoping”.

Alternative hint: The is like an Euler product for the sum $\sum_{n\geq 0} 2^{-n}$ with a binary expansion of $n$.

HINT: Multiply $1-\frac{1}{2^1}$ at the front.

- Real Analysis: Continuity of a Composition Function
- Multivariable calculus – Implicit function theorem
- Dilogarithm integral $\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\, dt$
- Properties of absolutely continuous functions
- Intersection of a properly nested sequence of convex sets , if nonempty and bounded , can never be open?
- Polish Spaces and the Hilbert Cube
- Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$
- Is the pointwise maximum of two Riemann integrable functions Riemann integrable?
- How do you prove the domain of a function?
- Visualizing $\cap_{i = 1}^\infty A_i = (\cup_{i = 1}^\infty A_i^c)^c$
- Cogroup structures on the profinite completion of the integers
- Ring homomorphism from $\mathbb Z$ to $\mathbb Z$ is always identity or $0$
- Finite order endomorphisms
- When can you simplify the modulus? ($10^{5^{102}} \text{ mod } 35$)
- Construction of an Irreducible Module as a Direct Summand