Oct 21, 2009.
Taylor Expansion.
A function f (x) that has n derivatives can be written as
1
f (x) = f (a) + f (1) (a)(x a) + f (2) (a)(x a)2 + + f (n) (a)(x a)n + Rn (x)
2
= P n ( x a ) + R n ( x)
How
Assignment 11. Due Dec 2, 2003.
Q1. Consider the space 1 of sequences a = cfw_an : n 1 such that
The distance between two sequences a and b is dened as
n=1
|an | = a < .
|an bn |
d(a, b) =
n=1
Show th
Assignment 8. Due Nov 11, 2003
Q 1. For p 1, lp is the space of sequences a = cfw_an with
|an |p < .
1
Check that a p = ( n |an |p ) p is a norm on lp and lp is complete under
this norm.
d
Q 2. If <
Oct 14, 2009.
Power Series.
a n xn = a 0 + a 1 x + a 2 x2 + + a n xn +
f ( x) =
n
It may not converge. It always converges when x = 0. If it converges for some value say
x = x0 , then an xn 0. Theref
If we have two nonnegative functions f (x) and g (x) and we compare their behavior as
x , i.e for large values of x, there are many possibilities. f (x) can get much larger
than g (x) which means
g (
1. A monotone class C is one with the the property that if either An A or An A and
An C for every n then A C . A eld F is a class that is closed under nite unions and
complementation. It follows then
Assignment 7. Due Nov 4, 2003
Q 1. Consider the following function d(x, y ) dened for irrational real
numbers E R.
d(x, y ) = |x y | +
j =1
1
1
1 | (xrj ) (yrj )|
1
1
2j 1 + | (xrj ) (yrj ) |
where cf
Assignment 3. Due September 30.
1. Given a continuous function f (x) on a closed subset C [0, 1] dene an extension of f
from C to [0, 1] as follows: If x C then there is an interval (a, b) such that a
Assignement 2. Due September 23, 2003
1. Let cfw_xn : n 1 be a sequence of points in [0, 1] and cfw_pn : n 1 a sequence of positive
numbers such that
m=
pn <
n
Show that
(A) =
pn
n:xn A
denes a set f
Assignment 9. Due Nov 18, 2003
Q 1. If (X, B, µ) is a ﬁnite measure space , say for instance µ(X) = 1, then
show that
Lp (X, B, µ) ⊂ Lp (X, B, µ)
for p ≥ p ≥ 1.
Q 2. For a function f ∈ ∩p≥1 Lp (X, B,
Model Questions.
Denite and Indenite Integration
1.
1
x dx
0
1 x
2.
1
x
dx
0
1 x
3.
1
0
x ex dx
4.
x sin x dx
5.
sin2 2x dx
6.
ex sin xdx
7.
dx
0 100+x2
8. Why is
2
0
f (sin x)dx =
9. Can you use it
Assignment 5. Due Oct 14, 2003
1. Let f (x) 0 be a continuous function on [0, 1]. We saw that if we dene f (x) by
x+h
1
|h|1 2h
f (x) = sup
f (y )dy
xh
then for some constant C , independent of f
[x :
Assignment 5. Due Oct 21, 2003
Q 1. We say that F is absolutely continuous in [0, 1] if for any given
> 0 such that
> 0, there exists a
N
|F (bj ) F (aj )|
j =1
when ever the intervals [aj , bj ] [0
Assignment 4. Due Oct 7, 2003.
Construct the Lebesgue measure (area) on the Borel subsets of the square cfw_(x, y ) : 0
x 1, 0 y 1 along the foloowing steps.
1. Start with Borel rectangles, i.e. sets