Intereting Posts

Equivalent Definitions of the Operator Norm
“Nice” well-orderings of the reals
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Parametrisation of the surface a torus
Sequence Notation — Which brackets to use?
$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$
“Pairwise independent” is weaker that “independent”
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Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ with $f(0,0)=0$ is a continuous function using epsilon-delta.
Two-Point boundary value problem
Weekend birthdays
Proving formula involving Euler's totient function
Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
How much weight is on each person in a human pyramid?
Is π equal to 180$^\circ$?

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

$$

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

$$

- Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
- Calculating probability with n choose k
- What is the best way to solve discrete divide and conquer recurrences?
- Coloring dots in a circle with no two consecutive dots being the same color
- On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$
- Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
- What are BesselJ functions?
- Finding number of functions from a set to itself such that $f(f(x)) = x$
- Inductive Proof that $k!<k^k$, for $k\geq 2$.
- Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$

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.

- Diagonalisable matrices question
- Evaluating $\sum\limits_{k=1}^{\infty}\frac1{(3k+1)(3k+2)}$
- Change of Basis Confusion
- Subjects studied in number theory
- A classic exponential inequality: $x^y+y^x>1$
- Expected area of the intersection of two circles
- Difference between Deformation Retraction and Retraction
- if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$
- Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
- If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
- Constructiveness of Proof of Gödel's Completeness Theorem
- Example of linear parabolic PDE that blows up
- Is $\sum\frac1{p^{1+ 1/p}}$ divergent?
- How to prove this result involving the quotient maps and connectedness?
- Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)