for the irrational number ln 2
≈
0
.
6931. This series was known and used by Euler.
Although we can integrate uniformly convergent sequences, we cannot in gen-
eral differentiate them. In fact, it’s often easier to prove results about the conver-
gence of derivatives by using results about the convergence of integrals, together
with the fundamental theorem of calculus. The following theorem provides suffi-
cient conditions for
f
n
→
f
to imply that
f
′
n
→
f
′
.
Theorem 1.64.
Let
f
n
: (
a, b
)
→
R
be a sequence of differentiable functions whose
derivatives
f
′
n
: (
a, b
)
→
R
are integrable on (
a, b
). Suppose that
f
n
→
f
pointwise
and
f
′
n
→
g
uniformly on (
a, b
) as
n
→ ∞
, where
g
: (
a, b
)
→
R
is continuous. Then
f
: (
a, b
)
→
R
is continuously differentiable on (
a, b
) and
f
′
=
g
.
Proof.
Choose some point
a < c < b
.
Since
f
′
n
is integrable, the fundamental
theorem of calculus, Theorem 1.45, implies that
f
n
(
x
) =
f
n
(
c
) +
integraldisplay
x
c
f
′
n
for
a < x < b.
Since
f
n
→
f
pointwise and
f
′
n
→
g
uniformly on [
a, x
], we find that
f
(
x
) =
f
(
c
) +
integraldisplay
x
c
g.
Since
g
is continuous, the other direction of the fundamental theorem, Theo-
rem 1.50, implies that
f
is differentiable in (
a, b
) and
f
′
=
g
.
square
In particular, this theorem shows that the limit of a uniformly convergent se-
quence of continuously differentiable functions whose derivatives converge uniformly
is also continuously differentiable.
The key assumption in Theorem 1.64 is that the derivatives
f
′
n
converge uni-
formly, not just pointwise; the result is false if we only assume pointwise convergence
of the
f
′
n
. In the proof of the theorem, we only use the assumption that
f
n
(
x
) con-
verges at a single point
x
=
c
. This assumption together with the assumption that
f
′
n
→
g
uniformly implies that
f
n
→
f
pointwise (and, in fact, uniformly) where
f
(
x
) = lim
n
→∞
f
n
(
c
) +
integraldisplay
x
c
g.
Thus, the theorem remains true if we replace the assumption that
f
n
→
f
pointwise
on (
a, b
) by the weaker assumption that lim
n
→∞
f
n
(
c
) exists for some
c
∈
(
a, b
).
This isn’t an important change, however, because the restrictive assumption in the
theorem is the uniform convergence of the derivatives
f
′
n
, not the pointwise (or
uniform) convergence of the functions
f
n
.
The assumption that
g
= lim
f
′
n
is continuous is needed to show the differ-
entiability of
f
by the fundamental theorem, but the result result true even if
g
isn’t continuous. In that case, however, a different (and more complicated) proof
is required.

40
1. The Riemann Integral
1.9.2. Pointwise convergence.
On its own, the pointwise convergence of func-
tions is never sufficient to imply convergence of their integrals.
Example 1.65.
For
n
∈
N
, define
f
n
: [0
,
1]
→
R
by
f
n
(
x
) =
braceleftBigg
n
if 0
< x <
1
/n,
0
if
x
= 0 or 1
/n
≤
x
≤
1
.
Then
f
n
→
0 pointwise on [0
,
1] but
integraldisplay
1
0
f
n
= 1
for every
n
∈
N
. By slightly modifying these functions to
f
n
(
x
) =
braceleftBigg
n
2
if 0
< x <
1
/n,
0
if
x
= 0 or 1
/n
≤
x
≤
1
,
we get a sequence that converges pointwise to 0 but whose integrals diverge to
∞
.

#### You've reached the end of your free preview.

Want to read all 55 pages?

- Winter '12
- JamesBremer
- Riemann integral, sup f