An efficient determination of the coefficients in the Chudnovskys’ series for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}

In 1914, Srinivasa Ramanujan published several hypergeometric series for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document}. One of these series was used by Bill Gosper in 1985 in a world-record-computation of π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}. Shortly after this, the Chudnovskys found a faster series for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document} based on the largest Heegner number and the Borweins proved Ramanujan’s series. Lately, the Chudnovskys’ series has often been used in practice to calculate digits of π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}, it reads: 640320312π=∑n=0∞6n!3n!n!313591409+545140134n-6403203n.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\sqrt{640320^3}}{12 \pi } = \sum _{n=0}^\infty \frac{\left( 6n\right) !}{\left( 3n\right) !\left( n!\right) ^3}\,\frac{13591409+ 545140134 n}{\left( -640320^3\right) ^n}. \end{aligned}$$\end{document}In this paper, we calculate the coefficients in two of Ramanujan’s series for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document} and those in the Chudnovskys’ series. For our calculation, we don’t require special software packages, but only the Fourier expansions of the Eisenstein series with a precision of ≈20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 20$$\end{document} decimals. We also prove the exactness of our calculations by proving that the values of certain non-holomorphic modular functions are algebraic integers. Our proof uses the division values of the Weierstraß ℘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\wp $$\end{document} function.