DPQ (version 0.5-3)

Bern: Bernoulli Numbers

Description

Return the \(n\)-th Bernoulli number \(B_n\), (or \(B_n^+\), see the reference), where \(B_1 = + \frac 1 2\).

Usage

Bern(n, verbose = getOption("verbose", FALSE))

Value

The number \(B_n\) of type numeric.

A side effect is the caching of computed Bernoulli numbers in the hidden environment

.bernoulliEnv.

Arguments

n

integer, \(n \ge 0\).

verbose

logical indicating if computation should be traced.

Author

Martin Maechler

References

https://en.wikipedia.org/wiki/Bernoulli_number

See Also

Bernoulli in Rmpfr in arbitrary precision via Riemann's \(\zeta\) function.

The next version of package gmp is to contain BernoulliQ(), providing exact Bernoulli numbers as big rationals (class "bigq").

Examples

(B.0.10 <- vapply(0:10, Bern, 1/2))
## [1]  1.00000000 +0.50000000  0.16666667  0.00000000 -0.03333333  0.00000000
## [7]  0.02380952  0.00000000 -0.03333333  0.00000000  0.07575758
if(requireNamespace("MASS")) {
  print( MASS::fractions(B.0.10) )
  ## 1  +1/2   1/6    0  -1/30     0  1/42     0 -1/30     0  5/66
}