
6-2 Diff erential Equations
Power series
An infi nite series of the form
0010
0
() ()
m
m
m
axx a ax x
∞
=
−=+−+
∑
2
20
()ax x−+
where
012
,,,aaa are real constants is called a power
series. These constants are called the coeffi cients of the series.
0
x
is a
constant and is called the centre of the series, and x is a real variable.
For the choice
0
0,x = we obtain the power series
2
01 2
0
m
m
m
ax a ax ax
∞
=
=+ + +
∑
(6.1)
with the origin as its centre.
Convergence interval
If there exists a positive real number R such that for all x in
(,)IxRxR=− +
the series Eq. (6.1) is convergent, then I is called
the interval of convergence for the series Eq. (6.1) and R is called ...