I design an educational program to demonstrate convergence rate in computation of $\pi$ by using the Machin like formula. According to Weisstein, the Hwang’s equation (see eq. (32) in http://mathworld.wolfram.com/Machin-LikeFormulas.html): $$\begin{align} \frac{1}{4}\pi = & 183{{\cot }^{-1}}239+32{{\cot }^{-1}}1023-68{{\cot }^{-1}}5832 \\ & +12{{\cot }^{-1}}110443-12{{\cot }^{-1}}4841182-100{{\cot }^{-1}}6826318 \\ \end{align}$$ is the most efficient Machin like formula to compute pi. I used three series for the inverse tangent ($\cot^{-1}x = \tan^{-1}\frac{1}{x}$): $$\tag{1}\tan^{-1}x=\sum\limits_{n=1}^{\infty }{\frac{{{\left( -1 \right)}^{n}}{{x}^{2n+1}}}{2n+1}},$$

$$\tag{2}\tan^{-1}x=\sum\limits_{n=1}^{\infty }{\frac{{{2}^{2n}}{{\left( n! \right)}^{2}}}{\left( 2n+1 \right)!}\frac{{{x}^{2n+1}}}{{{\left( 1+{{x}^{2}} \right)}^{n+1}}}},$$

$$\tag{3}\tan^{-1}x=2\sum\limits_{n=1}^{\infty }{\frac{1}{2n-1}\frac{{{a}_{n}}\left( x \right)}{a_{n}^{2}\left( x \right)+b_{n}^{2}\left( x \right)}}, $$ where $$\begin{align} & {{a}_{1}}\left( x \right)=2/x, \\ & {{b}_{1}}\left( x \right)=1, \\ & {{a}_{n}}\left( x \right)={{a}_{n-1}}\left( x \right)\left( 1-4/{{x}^{2}} \right)+4{{b}_{n-1}}\left( x \right)/x, \\ & {{b}_{n}}\left( x \right)={{b}_{n-1}}\left( x \right)\left( 1-4/{{x}^{2}} \right)-4{{a}_{n-1}}\left( x \right)/x. \\ \end{align}$$

Eq. (1) is the Maclaurin series, eq. (2) is the Euler’s series, eq. (3) is from the paper Abrarov & Quine arXiv:1706.08835. It appears to be that eqs. (2) and (3) are more rapid than eq. (1). I want to ask you two questions. Is eq. (3) more efficient than eq. (2)? Are there other series for the inverse tangent with rapid convergence?