Apéry's constant: Difference between revisions
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1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 |
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 |
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First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places: |
First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places): |
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1.202056<span style=color:red;>4036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111<span> |
1.202056<span style=color:red;>4036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111<span> |
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First 158 terms of Markov / Apéry representation truncated to 100 decimal places |
First 158 terms of Markov / Apéry representation truncated to 100 decimal places: |
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1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 |
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 |
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Revision as of 02:05, 24 February 2023
Apéry's constant is the sum of the reciprocals of the positive cubes.
That is, it is defined as the number where ζ is the Riemann zeta function.
Approximately equal to:
- 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
This constant was known and studied by many early mathematicians, but was not named until 1978, after Roger Apéry who was first to prove it was an irrational number.
is easy to calculate, but it converges very slowly. The first 1000 terms are only accurate to 6 decimal places.
There have been many fast convergence representations developed / discovered that generate correct digits much more quickly.
One of the earliest, discovered in the early 1800s by A. Markov and later widely published by Apéry is:
Much better than direct calculation of , but still only yielding about .63 correct digits per iteration.
Several even faster converging representions are available. The fastest known to date, yielding about 5.04 correct digits per term, is by Sebastian Wedeniwski.
- Task
- Show the value of Apéry's constant calculated at least three different ways.
- Show the value of at least the first 1000 terms of direct calculation truncated to 100 decimal digits.
- Show the value of the first 158 terms of Markov / Apéry representation truncated to 100 decimal digits.
- Show the value of the first 20 terms of Wedeniwski representation truncated to 100 decimal digits.
- See also
Raku
sub postfix:<!> (Int $n) { (constant f = 1, |[\×] 1..*)[$n] }
say 'Actual value to 100 decimal places:';
say '1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581';
say "\nFirst 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):";
say (1..1000).map({FatRat.new: 1, .³}).sum.substr: 0, 102;
say "\nFirst 158 terms of Markov / Apéry representation truncated to 100 decimal places:";
say (5/2 × (1..158).map( -> \k { (-1)**(k-1) × FatRat.new: k!², ((2×k)! × k³) } ).sum).substr: 0, 102;
say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";
say (1/24 × ((^20).map: -> \k {
(-1)**k × FatRat.new: (2×k+1)!³ × (2×k)!³ × k!³ × (126392×k⁵ + 412708×k⁴ + 531578×k³ + 336367×k² + 104000×k + 12463), (3×k+2)! × (4×k+3)!³
}).sum).substr: 0, 102;
- Output:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places): 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
First 158 terms of Markov / Apéry representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 20 terms of Wedeniwski representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581