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.

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