8.6. POWER SERIES
525
to find a power series expansion for f ′ (x).
(b) Observe that f (x) and f ′ (x) have familiar Taylor series. What familiar functions
are these? What known relationship does our work demonstrate?
(c) What is the series expansion for f ′′ (x)? What familiar function is f ′′ (x)?
⊳
It turns out that our work in Activity 8.29 holds more generally. The corresponding
theorem, which we will not prove, states that we can differentiate a power series for a
function f term by term and obtain the series expansion for f ′ , and similarly we can
integrate a series expansion for a function f term by term and obtain the series expansion
for
f (x) dx. For both, the radius of convergence of the resulting series is the same
as the original, though it is possible that the convergence status of the resulting series
may differ at the endpoints. The formal statement of the Power Series Differentiation and
Integration Theorem follows.
Power Series Differentiation and Integration Theorem. Suppose f (x) has a
power series expansion
f (x) =
∞
k=0
c k x
k
so that the series converges absolutely to f (x) on the interval −r < x < r. Then, the
power series
∞
k=1
kc k x
k−1 obtained by differentiating the power series for f (x) term
by term converges absolutely to f ′ (x) on the interval −r < x < r. That is,
f
′ (x) =
∞
k=1
kc k x
k−1
, for |x| < r.
Similarly, the power series
∞
k=0
c k
x k+1
k + 1
obtained by integrating the power series for
f (x) term by term converges absolutely on the interval −r < x < r, and
f (x) dx =
∞
k=0
c k
x k+1
k + 1
+ C, for |x| < r.
This theorem validates the steps we took in Activity 8.30. It is important to note
that this result about differentiating and integrating power series tells us that we can
differentiate and integrate term by term on the interior of the interval of convergence, but
it does not tell us what happens at the endpoints of this interval. We always need to check
what happens at the endpoints separately. More importantly, we can use use the approach
525
to find a power series expansion for f ′ (x).
(b) Observe that f (x) and f ′ (x) have familiar Taylor series. What familiar functions
are these? What known relationship does our work demonstrate?
(c) What is the series expansion for f ′′ (x)? What familiar function is f ′′ (x)?
⊳
It turns out that our work in Activity 8.29 holds more generally. The corresponding
theorem, which we will not prove, states that we can differentiate a power series for a
function f term by term and obtain the series expansion for f ′ , and similarly we can
integrate a series expansion for a function f term by term and obtain the series expansion
for
f (x) dx. For both, the radius of convergence of the resulting series is the same
as the original, though it is possible that the convergence status of the resulting series
may differ at the endpoints. The formal statement of the Power Series Differentiation and
Integration Theorem follows.
Power Series Differentiation and Integration Theorem. Suppose f (x) has a
power series expansion
f (x) =
∞
k=0
c k x
k
so that the series converges absolutely to f (x) on the interval −r < x < r. Then, the
power series
∞
k=1
kc k x
k−1 obtained by differentiating the power series for f (x) term
by term converges absolutely to f ′ (x) on the interval −r < x < r. That is,
f
′ (x) =
∞
k=1
kc k x
k−1
, for |x| < r.
Similarly, the power series
∞
k=0
c k
x k+1
k + 1
obtained by integrating the power series for
f (x) term by term converges absolutely on the interval −r < x < r, and
f (x) dx =
∞
k=0
c k
x k+1
k + 1
+ C, for |x| < r.
This theorem validates the steps we took in Activity 8.30. It is important to note
that this result about differentiating and integrating power series tells us that we can
differentiate and integrate term by term on the interior of the interval of convergence, but
it does not tell us what happens at the endpoints of this interval. We always need to check
what happens at the endpoints separately. More importantly, we can use use the approach