Graphics Reference
In-Depth Information
The blancmange function is continuous everywhere and
differentiable nowhere. A self-similar function like the
blancmange function is equally bumpy on every interval.
Summary
-
Differentiation and the
M
-test
T
heorem
If
qn 34
) converges
pointwiseto thefunction
f
:
A
R;
(ii) each of the functions
f
(i) the sequence of functions (
f
has a continuous
derivative on its domain
A
;
(iii) the sequence (
f
) converges uniformly to
:
A
R;
then
f
and (
f
) converges uniformly to
f
on
A
.
The Weierstrass M
-
test
qn 36
A sequence of functions (
f
) is defined by the
partial sums of a series:
f
(
x
)
u
(
x
).
If there exist real numbers
M
such that
(i)
u
(
x
)
M
, and
M
(ii)
is convergent,
) is uniformly convergent.
Theorem
A power series is uniformly convergent on any
qn 40 closed interval inside its circle of convergence.
Theorem
If
f
(
x
)
then (
f
a
x
has radius of convergence
R
,
qns 41, 42
then
f
(
x
)
f
a
x
n
1
na
x
and
when
x
R
.
The Binomial Theorem
a
n
qn 45
(1
x
)
x
, for all real
a
,
provided
1
x
1.
Historical note
Newton and his contemporaries differentiated and integrated power
series, term by term, without reference to their circle of convergence.
