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.

Diverging series, but only while it is not diverging - to get pi much faster

Ramanujan's formulas - again, faster convergence

Wham. It happens so fast. How?

Worksheet Questions

• Implement (2) methods of calculating pi using a for loop.

• For each method implemented, calculate pi using 100, 1k, and 10k terms in the series.

• Calculate the absolute error and clock time of each calculation.

• Create a plot of degree of accuracy versus number of terms. Use a log-log plot.

• Why does the plot from the previous question require log scales?

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

Nice history on Euler and summations to estimate pi: https://www.math.nmsu.edu/~davidp/euler2k2.pdf

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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