Q2.4.4The Madhava series

Use a for loop to estimate $\pi$ from the first 20 terms of the Madhava series:

\pi = \sqrt{12} \left( 1 - \frac{1}{3\cdot 3} + \frac{1}{5\cdot 3^2} - \frac{1}{7\cdot 3^3} + \cdots \right).

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The following calculates $\pi$ to 10 decimal places.

>>> import math
>>> pi = 0
>>> for k in range(20):
...     pi += pow(-3, -k) / (2*k+1)
...
>>> pi *= math.sqrt(12)
>>> print('pi = ', pi)
pi =  3.1415926535714034
>>> print('error = ', abs(pi - math.pi))
error =  1.8389734179891093e-11

Note that the built-in pow(x, j) is equivalent to (x)**j.