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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