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Ali Shadhar

I am a Bachelor's degree holder majored in physics. I like advanced calculus but my real interest goes to logarithmic/ polylogarithmic integrals and harmonic series.

I wrote a book about the harmonic series (An Introduction To The Harmonic Series And Logarithmic Integrals) and its available on Amazon.

Here is my best of on MSE:

$\bullet$ A simple proof of $\sum_{n=1}^\infty \frac{H_n}{n^42^n}$ and by only real methods.

$\bullet$ A proof of $\sum_{n=1}^\infty \frac{H_n}{n^32^n}$ based on a simple equality.

$\bullet$ Evaluation of $\int_0^1\frac{\arctan x}{x}\ln(\frac{1+x^2}{(1-x)^2})\ dx$ by using harmonic series.

$\bullet$ An advanced integral.

$\bullet$ The challenging integral $\int_0^1\frac{\arctan(x)\ln x}{1+x}dx$.

$\bullet$ Two powerful alternating Euler sums.

$\bullet$ Heavy Euler sum.

$\bullet$ Binomial sum.

$\bullet$ Challenging sum $\sum_{n=1}^\infty \frac{(-1)^nH_n^3}{n+1}$.

$\bullet$ Cornel's integral.

$\bullet$ Calculating logarithmic integrals without using the derivatives of Beta function.

$\bullet$ A group of important generating functions involving harmonic number.

$\bullet$ An easy way to calculate $\sum_{n=1}^\infty \frac{(-1)^nH_n}{n^4}$.

$\bullet$ A shortcut to compute $\int_0^{\pi/2} \frac{t \ln (1-\sin{t})}{\sin t}\ dt$.

$\bullet$ How to prove $\zeta(3)=2\sum_{n=1}^\infty\frac{H_n}{n}\left[\frac1{4^n}{2n\choose n}\left(H_{2n}-H_n-\frac1{2n}-\ln2\right)+\frac1{2n}\right]$.

$\bullet$ Three birds with one stone.

$\bullet$ The taylor series of $\frac{\ln^4(1-x)}{1-x}$.

$\bullet$ Generalization of $\sum_{n=1}^\infty \frac{H_n}{n^q}$ for odd $q$.

$\bullet$ $\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$ and $\sum_{n=1}^\infty\frac{H_n^{(2)}}{n^32^n}$

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