Convergence of Pi: Interactive Post

In an earlier post, I discussed how Pi can be expressed as a summation.

$$4 \sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{2 n-1}=4 \left(1 \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\text{…}\right)=\pi$$

I thought I would create some embedded Mathematica applets that show how the terms of the series converge to π. To start out, let’s look at how the summation converges as we add more an more terms.

Notice how the approximation jumps between overestimating and underestimating π.

The series gradually converges to π as more terms are added.

You can try evaluating the summation to different numbers of terms with this interactive tool.

Author: Wade

Materials scientist, lab manager, and educator based in the Seattle area.