Worksheets/Infinite Series Convergence

In this worksheet we study the convergence behavior of the series:

$ \sum_{i=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6} $

Explore a couple of methods of calculating pi, compare the results of a few methods, compare the computational cost. Push the limits.

Incorporate some kind of timing, I don't know.

Variables: number of terms, amount of time, and amount of accuracy.

Implement the following methods of calculating Pi using a for loop.

For each method implemented, calculate Pi using 100, 1,000, and 10,000 terms in the series.

For each calculation, calculate the absolute error and calculate the amount of time the calculation took.

References

Basel problem: https://en.wikipedia.org/wiki/Basel_problem

Proving the series converges (multiple ways): https://www.youtube.com/watch?v=9euTxoCC8Hk

Background on other convergent series: https://plus.maths.org/content/infinite-series-surprises

Ways of calculating pi: http://pi3.sites.sheffield.ac.uk/tutorials/week-7

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Link to all worksheets idea list: Worksheets

Calc II:

• Archimedes: Don't Disturb my Circles Worksheets/Archimdes_Dont_Disturb_My_Circles

• Simpson's Rule: Worksheets/Simpsons_Rule

• Civil Engineering Road Planning: Worksheets/Civil_Engineering_Road_Planning

• Euler's Method and Circuits: Worksheets/Eulers_Method_Circuits

Calc III:

• Infinite Series: Worksheets/Infinite_Series_Convergence

• Partial derivatives: Worksheets/Van Der Waal Equation Critical Point