Newbetuts

Show $1+\frac{8q}{1-q}+\frac{16q^2}{1+q^2}+\frac{24q^3}{1-q^3}+\dots=1+\frac{8q}{(1-q)^2}+\frac{8q^2}{(1+q^2)^2}+\frac{8q^3}{(1-q^3)^2}+\dots$.

Ramanujan series Type $\sum _{k=1}^{\infty } \frac{\sinh (2 \pi k)}{2 \sqrt{2} \pi ^9 k^{11} (1-\cosh (2 \pi k))}$

Evaluate $\lim\limits_{n \to \infty} nx_n.$

Are these new series formulae for $\zeta(2)$?

Convergence of the function series $\sum \frac{n!}{(nx)^n}$ for $x<0$

A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

Calculate $\sum\limits_{i=0}^\infty(2^{2^{(-i)}}-1)$

An interesting series related to primes satisfying $\sum_n x_{nk} = 0$ for all $k$

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.

Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$

Evaluating $\sqrt{6+\sqrt{6+\cdots}}$

Is every sequence defined on $\mathbb{R}$ just a countable subset of $\mathbb{R}$?

Information on the sum $\sum_{n=1}^\infty \frac{\log n}{n!}$

convergence of weighted average

Where did the negative answer come from in the continued fraction $1+\frac{1}{1+1/(1+\dots)}$?

convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$

New posts in sequences-and-series

Prove $\sum_{n=0}^\infty(-1)^n(\overline{H}_n-\ln2)^3=-\frac5{16}\zeta(3)$

Continued Fraction using all Perfect Squares

Ramanujan's infinite series for $\frac{x^3(3x-2)}{(2x-1)^3}$ for all positive integers $x$

On the density of a certain sequence of integers

Show $1+\frac{8q}{1-q}+\frac{16q^2}{1+q^2}+\frac{24q^3}{1-q^3}+\dots=1+\frac{8q}{(1-q)^2}+\frac{8q^2}{(1+q^2)^2}+\frac{8q^3}{(1-q^3)^2}+\dots$.

Ramanujan series Type $\sum _{k=1}^{\infty } \frac{\sinh (2 \pi k)}{2 \sqrt{2} \pi ^9 k^{11} (1-\cosh (2 \pi k))}$

Evaluate $\lim\limits_{n \to \infty} nx_n.$

Are these new series formulae for $\zeta(2)$?

Convergence of the function series $\sum \frac{n!}{(nx)^n}$ for $x<0$

A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

Calculate $\sum\limits_{i=0}^\infty(2^{2^{(-i)}}-1)$

An interesting series related to primes satisfying $\sum_n x_{nk} = 0$ for all $k$

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.

Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$

Evaluating $\sqrt{6+\sqrt{6+\cdots}}$

Is every sequence defined on $\mathbb{R}$ just a countable subset of $\mathbb{R}$?

Information on the sum $\sum_{n=1}^\infty \frac{\log n}{n!}$

convergence of weighted average

Where did the negative answer come from in the continued fraction $1+\frac{1}{1+1/(1+\dots)}$?

convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$