Telescoping sums and pi

You will add a series of inverse tangents term by term, write its telescoped closed form, and then use the same idea to compute $\pi$ with Machin's formula. The checker compares your functions on several values of $n$.

The idea. The terms $\tan^{-1}\dfrac{1}{r^2 + r + 1}$ add up to $\tan^{-1}(n + 1) - \dfrac{\pi}{4}$. Machin's formula is $\dfrac{\pi}{4} = 4\tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{239}$. It comes from the sum rule for inverse tangents, and it was used to compute $\pi$ by hand to many digits.

Your tasks.

1. Finish partial_sum(n). Add the terms for r from 1 to n, using a loop.
2. Finish telescoped(n). Return the closed form atan(n + 1) - pi / 4.
3. Finish machin_pi(). Return $4 \times \left(4\tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{239}\right)$.