52 lines
1.4 KiB
Text
52 lines
1.4 KiB
Text
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const n = (ARGV ? Num(ARGV[0]) : 50) # number of iterations
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define ℯ = Num.e
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define π = Num.pi
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define γ = Num.EulerGamma
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func display(r, t) {
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say "#{r}\terror: #{ '%.0g' % abs(r - t) }"
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}
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# Original definition of the Euler-Mascheroni constant, due to Euler (1731)
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display(sum(1..n, {|n| 1/n }) - log(n), γ)
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# Formula due to Euler (best convergence)
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display(harmfrac(n) - log(n) - 1/(2*n) - sum(1..n, {|k|
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-bernoulli(2*k) / (2*k) / n**(2*k)
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}), γ)
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# Formula derived from the above formula of Euler,
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# using approximations of Bernoulli numbers.
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display(harmfrac(n) - log(n) - 1/(2*n) - sum(1..n, {|k|
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(-1)**k * 4 * sqrt(π*k) * (π * ℯ)**(-2*k) * k**(2*k) / (2*k) / n**(2*k)
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}), γ)
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# Euler-Mascheroni constant, involving zeta(n)
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display(1 - sum(2..(n+1), {|n|
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(zeta(n) - 1) / n
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}), γ)
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# Limit_{n->Infinity} zeta((n+1)/n) - n} = gamma
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display(zeta((n+1)/n) - n, γ)
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# Series due to Euler (1731).
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display(sum(2..(n+1), {|n|
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(-1)**n * zeta(n) / n
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}), γ)
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# Formula due to Euler in terms of log(2) and the odd zeta values
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display(3/4 - log(2)/2 + sum(1..n, {|n|
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(1 - 1/(2*n + 1)) * (zeta(2*n + 1) - 1)
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}), γ)
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# Formula due to Euler in terms of log(2) and the odd zeta values (VII)
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display(log(2) - sum(1..n, {|n|
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zeta(2*n + 1) / (2*n + 1) / 2**(2*n)
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}), γ)
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# Formula due to Vacca (1910)
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display(sum(1..n, {|n|
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(-1)**n * floor(log2(n)) / n
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}), γ)
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