Newbetuts

Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$.

Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous

Find a smooth function with prescribed moments

How to improve $\int_{0}^{\frac{\pi}{2}}x\left(\frac{\sin(nx)}{\sin(x)}\right)^{4}dx<\frac{\pi^{2}n^{2}}{4}$

Proving that if $f>0$ and $\int_E f =0$, then $E$ has measure $0$

If $\int_A f\,dm = 0$ for all $A$ having some fixed measure $C$, then $f = 0$ almost everywhere

Asymptotic difference between a function and its binomial average

Special subset of an Euclidean space

Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?

Abel summability of integrals

Is the set of all polynomials in $\log(x)$ dense in $L^2[0,1]$?

Infinite self-convolution for a function

Proving completeness of Nikodym Metric

Are there some strategies to prove a set has measure zero?

On a relation between volume of subsets of $\mathbb R^n$

Limit points of particular sets of real numbers.

Showing that $f(x)=e^{-x}$ is uniformly continuous on $[0,\infty)$

New posts in real-analysis

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Proper Measurable subgroups of $\mathbb R$

When $L^p \subset L^q$ for $p <q$.

Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$.

Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous

Find a smooth function with prescribed moments

How to improve $\int_{0}^{\frac{\pi}{2}}x\left(\frac{\sin(nx)}{\sin(x)}\right)^{4}dx<\frac{\pi^{2}n^{2}}{4}$

Proving that if $f>0$ and $\int_E f =0$, then $E$ has measure $0$

If $\int_A f\,dm = 0$ for all $A$ having some fixed measure $C$, then $f = 0$ almost everywhere

Asymptotic difference between a function and its binomial average

Special subset of an Euclidean space

Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?

Abel summability of integrals

Is the set of all polynomials in $\log(x)$ dense in $L^2[0,1]$?

Infinite self-convolution for a function

Proving completeness of Nikodym Metric

Are there some strategies to prove a set has measure zero?

On a relation between volume of subsets of $\mathbb R^n$

Limit points of particular sets of real numbers.

Showing that $f(x)=e^{-x}$ is uniformly continuous on $[0,\infty)$