Kata

Let us define a function fsuch as:

• (1) for k positive odd integer > 2 :

• (2) for k positive even integer >= 2 :

• (3) for k positive integer > 1 :

where |x| is abs(x) and Bk the k^th Bernoulli number.

f is not defined for 0, 1, -1. These values will not be tested.

Guidelines for Bernoulli numbers:

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

http://mathworld.wolfram.com/BernoulliNumber.html

https://www.codewars.com/kata/bernoulli-numbers-1

There is more than one way to calculate them. You can make Pascal triangle and then with the basic formula below generate all Bernoulli numbers.

1 + 2B = 0 ... gives ... B = - 1/2

1 + 3B + 3B = 0 ... gives ... B = 1/6

1 + 4B + 6B + 4B = 0 ... gives ... B = 0

1 + 5B + 10B + 10B + 5B = 0 ... gives ... B = - 1/30

... and so on

Task

Function series(k, nb) returns (1), (2) or (3) where k is the k parameter of f and nb the upper bound in the summation of (1). nb is of no use for (2) and (3) but for the sake of simplicity it will always appear in the tests even for cases (2) and (3) with a value of 0.

Examples
S(2, 0) = 1.644934066848224....
S(3, 100000) = 1.20205690310973....
S(4, 0) = 1.08232323371113.....
S(-5, 0) = -0.003968253968....
S(-11, 0) = 0.02109279609279....

Notes

• For Java, C#, C++: k should be such as -27 <= k <= 20; otherwise -30 <= k <= 30 and be careful of 32-bit integer overflow.
• Translators are welcome.