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I have a Bachelor's degree in physics and I enjoy 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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