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.

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