New posts in real-analysis

An Euler type sum: $\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n\cdot 4^n}{2n \choose n}$, where $H_n^{(2)}=\sum\limits_{k=1}^{n}\frac{1}{k^2}$

real-analysis calculus sequences-and-series summation harmonic-numbers

Calculate the sum $S_n = \sum\limits_{k=1}^{\infty}\left\lfloor \frac{n}{2^k} + \frac{1}{2}\right\rfloor $

real-analysis discrete-mathematics summation ceiling-and-floor-functions

Are there continuous functions $f,g:\mathbb{R}\longrightarrow\mathbb{R}$ such that for any $x$, $f(g(x))=\sin x$ and $g(f(x))=\cos x$?

real-analysis functions functional-equations

Proof of a few equations involving $\int_{\alpha}^{\infty}\frac{1}{t\left(e^{t}\pm1\right)}dt$

real-analysis integration closed-form

Spivak Chapter 10, Problem 29 : why is this true?

real-analysis calculus

Computing $\sum_{n=1}^\infty\frac{2^{2n}H_{n+1}}{(n+1)^2{2n\choose n}}$

real-analysis integration sequences-and-series binomial-coefficients harmonic-numbers

Find the sum: $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$

real-analysis calculus sequences-and-series summation power-series

Is every "almost everywhere derivative" Henstock–Kurzweil integrable?

real-analysis integration analysis derivatives gauge-integral

Uniform semi-continuity

How much a càdlàg (i.e., right-continuous with left limits) function can jump?

What is the minimum value of $a$ such that $x^a \geq \ln(x)$ for all $x > 0$?

calculus real-analysis

Differentiability and decay of magnitude of fourier series coefficients

real-analysis fourier-series

Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

real-analysis functional-analysis fourier-analysis banach-algebras

Given a Borel set $B$ prove: for every $\epsilon$, $\exists$ compact and closed sets and a continuous $\phi$...

real-analysis measure-theory lebesgue-integral

Equivalence of the Lebesgue integral and the Henstock–Kurzweil integral on nonnegative real functions

real-analysis integration measure-theory lebesgue-integral gauge-integral

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.

real-analysis algebra-precalculus trigonometry pi

Any elementary proof for Euler's product formula for sine [duplicate]

real-analysis education

Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert_{\infty}$ is complete. [duplicate]

real-analysis measure-theory banach-spaces

Is Spivak wrong about this counterexample? $f$ integrable on $[-1,1]$, $F=\int_{-1}^xf$, $f$ differentiable at $0$, but $F'$ not continuous at $0$

real-analysis calculus limits

Prove that there exists a sequence $(x_n)$ such that $\sum_n a_n x_n$ diverges

real-analysis sequences-and-series functional-analysis alternative-proof