Initial data commit

Task/Eulers-constant-0.5772.../Sidef/eulers-constant-0.5772...-1.sidef

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+# 100 decimals of precision
+local Num!PREC = 4*100
+say Num.EulerGamma

Task/Eulers-constant-0.5772.../Sidef/eulers-constant-0.5772...-2.sidef

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