Gaussian Integral and A Pi Approximator

I remember how I was happy when I find the integration of the Gaussian function in my high school years. It was an instantaneous spark which make me realize that increasing the integration dimension may bring a solution. Moreover, one can find a series expansion for Pi number after evaluating the integral.

Gaussian Integral

What is the Gaussian integral: Finding the area under the Gaussian function. I = \int_{-\infty}^{+\infty}\, e^{-\alpha x^2}\,dx

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