292 CHAPTER 5 Series and Differential Equations
For most of the functions we’ve considered here, this is true, but the
general result can fail.^19 As a result, we shall adopt the notation
f(x) ∼∑∞
k=0f(k)(a)
n!(x−a)kto mean thatf(x)is represented by the power series
∑∞
k=0f(k)(a)
n!(x−a)k; in Subsection 3.2 we’ll worry about whether “∼” can be replaced
with “=”. First, however, we shall delve into some computations.
5.4.1 Computations and tricks
In this subsection we’ll give some computations of some Taylor and
Maclaurin series, and provide some interesting shortcuts along the way.
Example 1. Letf(x) = sinxand find its Maclaurin series expansion.
This is simple as the derivatives (atx= 0) are easy to compute
f(0)(x) = sin 0 = 0, f′(0) = cos 0 = 1, f′′(0) =−sin 0 = 0,f′′′(0) =−cos 0 =− 1 ,f(4)(0) = sin 0 = 0,and the pattern repeats all over again. This immediately gives the
Maclaurin series for sinx:
sinx∼x−x^3
3!+
x^5
5!−
x^7
7!+···=
∑∞
n=0(−1)nx^2 n+1
(2n+ 1)!.
(^19) As an example, consider the functionfdefined by setting
f(x) =
®e− 1 /x (^2) ifx 6 = 0
0 ifx= 0.
One can show that all derivatives offvanish atx= 0 and so cannot equal its Maclaurin series in
any interval aboutx= 0.