Intereting Posts

Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$
Generate a number with a die that has three 0s and three 1s
Unpublished Discoveries by Gauss that Were Later Rediscovered and Attributed to Other Mathematicians
How to determine if I'm talented enough to study math?
Bijection between an infinite set and its union of a countably infinite set
Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
Is bijection mapping connected sets to connected homeomorphism?
Minimum degree of a graph and existence of perfect matching
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff
Coloring Graph Problem
How many rolls do I need to determine if my dice are fair?
Will computers one day start creating and proving math for us?
can the emphasis on “smallest” in the monotone class theorem be ignored in applications?
Mathematics in French

I’ve been exploring functions that have a general form:

$$\sum_{k=0}^\infty{ a^{b^k} } \tag{1}$$

In particular, I’m now checking this equality, which seems to hold:

- Intuitive explanation of Cauchy's Integral Formula in Complex Analysis
- Fundamental group of projective plane is $C_{2}$???
- Intuitive Explanation of Morphism Theorem
- Geometric intuition for the tensor product of vector spaces
- What is the intuition behind the name “Flat modules”?
- Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements

$$2 \sum_{k=0}^\infty{ \left( \frac{1}{2^{2^k}} – \frac{1}{2^{2^k\cdot3-1}} \right) } = 5/6$$

I’m also in the process of finding more identities/equations, but I don’t want to reinvent the wheel.

So I’m wondering, **What is known about series of the form (1)?** I’m interested in this and anything related to “doubly exponential” series. I’d be extremely interested in any books or papers that anyone knows about.

- about a good book - Vector Calculus
- Does every continuous time minimal Markov chain have the Feller property?
- Help understanding Algebraic Geometry
- Is there an English translation of Diophantus's Arithmetica available?
- Reference request: Where is this trigonometric identity found?
- book for metric spaces
- Comparison theorem for systems of ODE
- Reference for Fredholm Integral Equations..
- How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?
- Prison problem: locking or unlocking every $n$th door for $ n=1,2,3,…$

The function

$$

f(z)=\sum_{k=0}^{\infty}z^{a^k}=z+z^a+z^{a^2}+z^{a^3}+\ldots,

$$

where $a$ is a positive integer, is analytic for $|z|<1$, equal to $0$ at $z=0$, and satisfies the functional equation

$$

f(z^a)=f(z)-z.

$$

For $a=2$, you have the additional fun property that

$$

f(z)+f(z^3)+f(z^5)+\ldots=\frac{z}{1-z}.

$$

- Primitive roots of unity
- Bound of a complex-valued function
- Chance of meeting in a bar
- a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$
- How does a branch cut define a branch?
- Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$
- Probability Brownian motion is positive at two points
- How to understand the combination formula?
- The integral of a closed form along a closed curve is proportional to its winding number
- Why free presentations?
- Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n – 1$ by induction
- Compact inclusion in $L^p$
- Research in plane geometry or euclidean geometry
- Different versions of Riesz Theorems
- Solve a problem on integration