608
B Review of Traditional Calculus
B k =
d k
dx k
x
e x − 1
x=0
.
It will also be fairly obvious that this is not necessarily a particularly practical means
for determining these numbers, especially for larger integers. A more practical
means for generating values for the B k can be obtained from a rearrangement of
Eq. (B.1.20) into the form
1
x
(e
x
− 1)
∞
k=0
B k x k
k!
= 1 ,
followed by expansion of e x , multiplication of the two resultant series expansions,
and utilization of the linear independence of the powers of x to evaluate the B k
coefficients sequentially, starting from k = 0. 3 This process readily gives B 0 = 1,
B 1 = −
1
2 , B 2 =
1
6 , B 3 = 0, B 4 = −
1
30 , B 5 = 0, B 6 =
1
42 , B 7 = 0, B 8 = −
1
30 ,
and so on. It turns out that B 2k+1 = 0, for k ≥ 1. Note also that Bernoulli numbers
may now readily be evaluated by either of the above methods upon utilizing modern
symbolic computation packages.
B.1.5 The Euler–Maclaurin Summation Formula
We follow here a heuristic treatment by Gautschi 4 to obtain the Euler–Maclaurin
summation formula by starting from the well-known Taylor series expression for a
function f (x), namely,
f (x) = f (a) + f
(a)(x − a) +
1
2!
f
(a)(x − a)
2
+ · · · .
For x near in value to a, we may write x = a + h, so that the Taylor series
expression for f (x) may be expressed in terms of the differential operator D ≡
d
da
and rearranged into the form
f (h + a) − f (a) =
∞
k=1
h k D k
k!
f (a) .
Substitution of x for a then produces a version of the Taylor expansion of f (x) in
operator format, i.e.,
3 See, e.g., G. Arfken, Mathematics for Physicists, 2nd Ed. (Academic Press, New York, 1970).
4 W. Gautschi, Numerical Analysis: An Introduction (Birkhäuser, Boston, 1997).