Newbetuts

Show that $\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$ [duplicate]

Show that $\exists \delta > 0, \forall x \in ]0,\pi[, \exists n \in \Bbb N, |\sin(xk^n)|\ge \delta$.

Find the rate of convergence of given sequence.

Underdog leading at least once in an infinite series of games

Least permutations needed to permute from decreasing order to increasing order

Closed form expression for the harmonic sum $\sum\limits_{n=1}^{\infty}\frac{H_{2n}}{n^2\cdot4^n}{2n \choose n}$

Decide whether $\sum_{n=1}^{\infty}(-1)^n\frac{nx}{1+n^4x^2}$ uniformly converges on $[0,\infty)$ or not

Prove the limit exists

Calculating sum of converging series $\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n}$

$\int_0^1\frac{\ln x\ln^2(1-x^2)}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\zeta(3)-2\pi\ln^32$

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Proving Holder's inequality for sums

Is this proof for the limit law of the product of converging sequences correct? [duplicate]

How can I derive what is $1\cdot 2\cdot 3\cdot 4 + 2\cdot 3\cdot 4\cdot 5+ 3\cdot 4\cdot 5\cdot 6+\cdots + (n-3)(n-2)(n-1)(n)$ ??

Sum of a rearranged alternating harmonic series, with three positive terms followed by one negative term

For any unbounded set of real numbers, is there a subset which almost coincides with a uniformly spread out set of points an infinite amount of times?

Evaluation of a dilogarithmic integral

New posts in sequences-and-series

Formula for some average

How I find limit of $P_n$

How do I manipulate the sum of all natural numbers to make it converge to an arbitrary number?

Show that $\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$ [duplicate]

Show that $\exists \delta > 0, \forall x \in ]0,\pi[, \exists n \in \Bbb N, |\sin(xk^n)|\ge \delta$.

Find the rate of convergence of given sequence.

Underdog leading at least once in an infinite series of games

Least permutations needed to permute from decreasing order to increasing order

Closed form expression for the harmonic sum $\sum\limits_{n=1}^{\infty}\frac{H_{2n}}{n^2\cdot4^n}{2n \choose n}$

Decide whether $\sum_{n=1}^{\infty}(-1)^n\frac{nx}{1+n^4x^2}$ uniformly converges on $[0,\infty)$ or not

Prove the limit exists

Calculating sum of converging series $\sum_{n=1}^{\infty}\frac{1-n}{9n^3-n}$

$\int_0^1\frac{\ln x\ln^2(1-x^2)}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\zeta(3)-2\pi\ln^32$

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Proving Holder's inequality for sums

Is this proof for the limit law of the product of converging sequences correct? [duplicate]

How can I derive what is $1\cdot 2\cdot 3\cdot 4 + 2\cdot 3\cdot 4\cdot 5+ 3\cdot 4\cdot 5\cdot 6+\cdots + (n-3)(n-2)(n-1)(n)$ ??

Sum of a rearranged alternating harmonic series, with three positive terms followed by one negative term

For any unbounded set of real numbers, is there a subset which almost coincides with a uniformly spread out set of points an infinite amount of times?

Evaluation of a dilogarithmic integral