Intereting Posts

Prove or disprove that $\exists a,b,c\in\mathbb{Z}_+\ \forall n\in\mathbb{Z}_+\ \exists k\in\mathbb{Z}_+\colon\ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n)$
How many edges does an undirected tree with $n$ nodes have?
Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems
Knight move variant: Can it move from $A$ to $B$
Antipodal mapping of the sphere
Prove the identity Binomial Series
What is the exact definition of a reflexive relation?
When does a line bundle have a meromorphic section?
proving $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$
Infinite dimensional intermediate subfields of an algebraic extension of an algebraic number field
Continuous bounded functions in $L^1$
Limit of a sequence that tends to $1/e$
Is $\mathbb{C}^2$ isomorphic to $\mathbb{R}^4$?
Why are punctured neighborhoods in the definition of the limit of a function?
Ordinary generating functions problem

Can this series be expressed in closed form, and if so, what is it?

$$

\sum_{n=1}^\infty\frac{1}{9^{n+1}-1}

$$

- Computing RSA Algorithm
- About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$
- What are BesselJ functions?
- Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$
- Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$
- Trouble understanding poset notation and Hasse Diagrams
- Prove that $\mathbb{|Q| = |Q\times Q|}$
- why $m$ power by $n$ equals sum of $n$ numbrs
- How to find inverse of 2 modulo 7 by inspection?
- Closed form of $\displaystyle\mathscr{R}=\int_0^{\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx$

Let us denote by $f$ the following function

$$ f(a):=\sum_{n=1}^\infty\frac{1}{a^{n+1}-1}. $$

Then

$$f(a) =

\sum_{n=1}^\infty\frac{1}{a^{n+1}-1}=

\frac{1}{1-a}-\sum_{n=1}^\infty\frac{1}{1-a^n}\stackrel{\left(\spadesuit\right)}{=}

\frac{1}{1-a}-\frac{\psi_{1/a}(1)+\ln(a-1)+\ln(1/a)}{\ln(a)},$$

where $\psi_q(z)$ denotes the $q$-polygamma function, and in $\left(\spadesuit\right)$ we used the equation $(4)$ from here.

$$

f(9) = \frac78 – \frac{\ln(8)}{\ln(9)} – \frac{\psi_{1/9}(1)}{\ln(9)} \approx 0.014045117662188129358728474369089\dots

$$

Note that $f(2)=\mathcal{C}_{\textrm{EB}}-1 \approx 0.60669515241529\dots,$ where $\mathcal{C}_{\textrm{EB}}$ is the Erdős–Borwein constant.

- All real numbers in $$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$
- Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?
- Residue at infinity (complex analysis)
- $\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.
- $\operatorname{Soc}(\operatorname{Aut}( G))$ is isomorphic to $G$, for $G$ a nonabelian, simple group.
- How to solve this confusing permutation problem related to arrangement of books?
- Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
- Heat equation with discontinuous sink and zero flux boundary conditions
- How many monotonically increasing sequences of natural numbers?
- A contradiction involving exponents
- Field and Algebra
- calculate the sum of an infinite series
- Counter-example: Cauchy Riemann equations does not imply differentiability
- Calculating alternating Euler sums of odd powers
- Covering map is proper $\iff$ it is finite-sheeted