Saturday, May 27, 2023

TI-Nspire CX II (CAS): How Many Terms Are Needed to Calculate π?

TI-Nspire CX II (CAS): How Many Terms Are Needed to Calculate π?



Every One Wants a Slice of π


There are many ways of calculating the π, usually apply a formula or a sequence many, many, many times to get as many digits as possible.  The file pitests.tns tests (see below) five algorithms:



Test 1:  Zeta Formula


π^2 / 6 = Σ( 1 / n^2, n=1 to ∞ )


Test 2:  Pi Squared Over Eight Sum


π^2 / 8 = Σ( (2 * n + 1)^(-2), n = 0 to ∞)


Test 3:  Half of Pi Product


π / 2 = Π( (2 * k)^2 / ((2 * k - 1) * (2 * k + 1)), k = 1 to ∞)


Test 4:  Fibonacci Number Sum


π / 2 = Σ( arctan((F_[2*k+1])^(-1)), k = 0 to ∞ )


The angle is radians.  F_[2*k+1] is the [2*k+1]th Fibonacci number.


Test 5:  Limit Sequence


π = lim (n → ∞) [ a^2 / n ]


where 

a0 = 1

a_[n+1] = a_[n] * (1 + (2 * n + 1)^(-1))


The goal is to determine how many terms are needed to obtain a certain amount of digits.  






Certain algorithms take longer than others.     Here are some results:


2 places:  3.14

3 places:  3.141



Test 1:  

2 decimal places:  600 terms

3 decimal places:  1611 terms 


The zeta function converges super slowly for small arguments.


Test 2:

2 decimal places:  199 terms

3 decimal places:  537 terms


Test 3:

2 decimal places:  493 terms

3 decimal places:  1325 terms


Test 4:

2 decimal places:  9 terms

3 decimal places: 10 terms


The Fibonacci method is the fastest and probably is the quickest way to build the expansion of π.


Test 5:

2 decimal places:  94 places

3 decimal places:  1929 places



Download the file pitests.tns here:  https://drive.google.com/file/d/1XqHtPnoKvqfmFLNWHQ-IayFz_lM5zZyT/view?usp=share_link



Source


"List of formulae involving π"  Wikipedia.   Edited on February 11, 2023.   Accessed on March 12, 2023.  https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80



Note:   On June and July 2023, regular posts will be on Saturdays only, which will include the next Carnival of Math.


Eddie



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