Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 1, 2010
Juris Stepr ns
a
MATH 2001
The Generalized Mean Value Theorem
Theorem
Let f and F be functions such that f and F are both continuous
on [a, b ] and F (x ) = 0 for all

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
October 22, 2010
Juris Stepr ns
a
MATH 2001
Continuity and the intermediate value
property
Theorem
If f is a continuous function on [a, b ] and f (a) A f (b ) then
there is some c bet

Recall the limit game
Recall that limx a f (x ) = T is the same as saying that Player I
has no winning strategy in the following game:
1
Player I choose
> 0.
2
Player II choose > 0.
3
Player I choose x in the interval (a , a ).
4
The referee then measures

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
October 4, 2010
Juris Stepr ns
a
MATH 2001
Solving differential equations with power
series
To solve the dierential equation
f (x ) + xf (x ) cos(x ) = 0
one can look for solutions of

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 29, 2010
Juris Stepr ns
a
MATH 2001
Theorem
Let cfw_fn be a sequence of functions such that
n=0
The series
n=0 fn (a)
converges to F (a).
There is an interval I around the p

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 24, 2010
Juris Stepr ns
a
MATH 2001
Definition
A sequence of functions cfw_fn converges pointwise to the
n=0
function f on A if limn fn (x ) = f (x ) for all x in A.
Definit

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 22, 2010
Juris Stepr ns
a
MATH 2001
Rearrangement of series
Theorem (A)
If an is absolutely convergent and cfw_bn is a
n=1
n=1
rearrangement of cfw_an (formally, : N N is a

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 15, 2010
Juris Stepr ns
a
MATH 2001
Ratio test example
The series
n=1
n!
nn
is convergent. By the ratio test
an+1
(n + 1)!nn
(n + 1)nn
nn
=
=
=
=
an
n!(n + 1)n+1
(n + 1)n+1
(

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 17, 2010
Juris Stepr ns
a
MATH 2001
The value of a series of functions fn can be dened in
n=1
various ways. The pointwise limit is the most obvious, but its
behaviour is unpr

Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
November 10, 2010
Juris Stepr ns
a
MATH 2001
Notation
The sum a0 + a1 + a2 + . . . + an will be denoted by n=0 aj .
j
More generally, the sum ak + ak +1 + ak +2 + . . . + an will be
d

Mid Term Examination November 5, 2010.
MATH2001
Attempt all questions. No aides are allowed. The time allowed for completion is 50 minutes.
(1) Let an be a real number for each natural number n. Explain the meaning of the following
statement:
a1 + a2 + a3

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