4.6. REPRESENTATION OF FUNCTIONS AS POWER SERIES 99
4.6. Representation of Functions as Power Series
We have already seen that a power series is a particular kind of
function. A slightly different matter is that sometimes a given function
can be written as a power series. We already know the example
1
1−x= 1 + x+x2+x3+···+xn+···=∞
X
n=0
xn(|x|<1)
Replacing xwith other expressions we may write other functions in the
same way, for instance by replacing xwith −2x2we get:
1
1 + 2x2= 1−2x2+4x4−8x6+···+(−1)n2nx2n+···=∞
X
n=0
(−1)n2nx2n,
which converges for |−2x2|<1, i.e., |x|<1/√2.
4.6.1. Differentiation and Integration of Power Series. Since
the sum of a power series is a function we can differentiate it and in-
tegrate it. The result is another function that can also be represented
with another power series. The main related result is that the deriv-
ative or integral of a power series can be computed by term-by-term
differentiation and integration:
4.6.1.1. Term-By-Term Differentiation and Integration. If the power
series P∞
n=0 cn(x−a)nhas radius of convergence R > 0 then the func-
tion
f(x) = ∞
X
n=0
cn(x−a)n
is differentiable on the interval (a−R, a +R) and
(1) f0(x) = ∞
X
n=0{cn(x−a)n}0=∞
X
n=1
ncn(x−a)n−1
(2) Zf(x)dx =∞
X
n=0 Zcn(x−a)ndx =C+∞
X
n=0
cn
(x−a)n+1
n+ 1
The radii of convergence of the series in the above equations is R.
Example: Find a power series representation for the function
f(x) = 1
(1 −x)2.
