Worksheets/Infinite Series Convergence: Difference between revisions
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\sum_{i=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6} | \sum_{i=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6} | ||
</math> | </math> | ||
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 | |||
[[Image:PythonHugeFactorials.png]] | |||
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= | =References= | ||
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Background on other convergent series: https://plus.maths.org/content/infinite-series-surprises | 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 | |||
=Flag= | =Flag= | ||
{{WorksheetFlag}} | {{WorksheetFlag}} | ||
Latest revision as of 07:42, 14 May 2016
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
Flag
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