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.

What is the cardinality of a set of all monotonic functions on a segment $[0,1]$?

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

the value of $\lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$

Show that if $f$ is integrable on $[a,b]$, then $|f|$ is also integrable.

Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$

Finding $\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$

Can we construct a function that has uncountable many jump discontinuities?

Is $(a,a]=\{\emptyset\}$?

Bounded function and second derivative implies bounded derivative.

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

How to show that the set of all Lipschitz functions on a compact set X is dense in C(X)?

Finite unions and intersections $F_\sigma$ and $G_\delta$ sets

Prove the limit exists

Proving that $8^x+4^x\geq 5^x+6^x$ for $x\geq 0$.

The Multiplication Operator $M_f: L^2(\mu) \to L^2(\mu)$ such that $M_f g = fg$ (Rudin)

New posts in real-analysis

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 $f(x)=\cos(x)$ is Lipschitz continuous function.

Arithmetic-quadratic mean and other "means by limits of means"

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.

What is the cardinality of a set of all monotonic functions on a segment $[0,1]$?

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

the value of $\lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$

Show that if $f$ is integrable on $[a,b]$, then $|f|$ is also integrable.

Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$

Finding $\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$

Can we construct a function that has uncountable many jump discontinuities?

Is $(a,a]=\{\emptyset\}$?

Bounded function and second derivative implies bounded derivative.

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

How to show that the set of all Lipschitz functions on a compact set X is dense in C(X)?

Finite unions and intersections $F_\sigma$ and $G_\delta$ sets

Prove the limit exists

Proving that $8^x+4^x\geq 5^x+6^x$ for $x\geq 0$.

The Multiplication Operator $M_f: L^2(\mu) \to L^2(\mu)$ such that $M_f g = fg$ (Rudin)