B.1 Derivatives
607
for the Maclaurin series expansion of a function f (x) about x = 0. You will also
recall that the Maclaurin series is just a special case of the Taylor series, for a = 0.
Those expansions most commonly encountered are the Maclaurin expansions of the
functions e x , sinx, cosx, sinhx, coshx, tan −1 x, and (1 − x) −1 . These expansions are
given by
e
x
= 1 + x +
x 2
2!
+ · · · =
∞
i=0
x i
i!
,
(B.1.19a)
sin x = x −
x 3
3!
+
x 5
5!
+ · · · =
∞
i=0
(−1) i x 2i+1
(2i + 1)!
,
(B.1.19b)
cos x = 1 −
x 2
2!
+
x 4
4!
+ · · · =
∞
i=0
(−1) i x 2i
(2i)!
,
(B.1.19c)
sinh x = x +
x 3
3!
+
x 5
5!
+ · · · =
∞
i=0
x 2i+1
(2i + 1)!
,
(B.1.19d)
cosh x = 1 +
x 2
2!
+
x 4
4!
+ · · · =
∞
i=0
x 2i
(2i)!
,
(B.1.19e)
tan
−1 x = x −
x 3
3
+
x 5
5
− · · · =
∞
i=0
(−1) i x 2i+1
2i + 1
,
(B.1.19f)
1
1 − x
= 1 + x + x
2
+ · · · =
∞
i=0
x
i .
(B.1.19g)
Note that, although the first five of these expansions hold for all values of x, the
latter two expansions are valid (i.e., converge) only for −1 ≤ x ≤ 1 for the inverse
tangent function, and −1 < x < 1 for the reciprocal of 1 − x.
Example (Bernoulli Numbers) an application of infinite series representation.
Bernoulli numbers, B k , are obtained by employing the infinite series representation of the function x/(e x − 1) to define them via
x
e x − 1
≡
∞
k=0
B k
k!
x
k .
(B.1.20)
It will be clear that the Bernoulli numbers may be obtained formally by carrying out
repeated differentiation of the two sides of this defining relation, followed by setting
x to zero, to give the B k as