Up to this point, we have shown several techniques for finding power series representations for functions. However, how do we know that these power series are unique? That is, given a function
f and a power series for
f at
a , is it possible that there is a different power series for
f at
a that we could have found if we had used a different technique? The answer to this question is no. This fact should not seem surprising if we think of power series as polynomials with an infinite number of terms. Intuitively, if
for all values
x in some open interval
I about zero, then the coefficients
c
n should equal
d
n for
We now state this result formally in
[link] .
Uniqueness of power series
Let
and
be two convergent power series such that
for all
x in an open interval containing
a . Then
for all
Proof
Let
Then
By
[link] , we can differentiate both series term-by-term. Therefore,
and thus,
Similarly,
implies that
and therefore,
More generally, for any integer
and consequently,
for all
□
In this section we have shown how to find power series representations for certain functions using various algebraic operations, differentiation, or integration. At this point, however, we are still limited as to the functions for which we can find power series representations. Next, we show how to find power series representations for many more functions by introducing Taylor series.
Key concepts
Given two power series
and
that converge to functions
f and
g on a common interval
I , the sum and difference of the two series converge to
respectively, on
I . In addition, for any real number
b and integer
the series
converges to
and the series
converges to
whenever
bx
m is in the interval
I .
Given two power series that converge on an interval
the Cauchy product of the two power series converges on the interval
Given a power series that converges to a function
f on an interval
the series can be differentiated term-by-term and the resulting series converges to
on
The series can also be integrated term-by-term and the resulting series converges to
on
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