1
CALCULUS I
DIFFERENTIAL CALCULUS
Mathematical Analysis I
1
CALCULUS I
DIFFERENTIAL CALCULUS
Mathematical Analysis I
2
2
3
2
Pythagoreans
a school
,
in some ways a brotherhood, and in
some ways a monastery.
Pythagoras of Samos
between 580 and 572 BC
(between 500 and 490 BC)
4
Zeno of Elea (490-430 BC): On the paradox of motion
5
Achilles
Tortoise
6
7
ISAAC NEWTON 1643 – 1727, England
8
Gottfried Leibniz, 1647-1716, Germany
9
Augustin Louis Cauchy, 1789 -1857, France
10
LIMIT OF A SEQUENCE
Lecture 1
11
A SEQUENCE
12
A SEQUENCE
13
,
x
x
0
...
,
1
,
0
n
,
x
,
,
x
,
x
F
x
0
1
n
n
1
n
r
x
x
n
1
n
n
1
n
x
q
x
Example 2
recursive definition
arithmetic sequence - common difference
geometric sequence - common ratio
14
15
n
1
2
1
a
n
n
)
(
is divergent
Example 4
16
Example 5 Take a calculator, set it to "radian mode" and enter the
number 1. Then, hit the function cosine over and over again. Analyse the
output of this experiment.
17
We graph the points (n, x
n
) on a plane
The points ‘converge’ to about 0.73
18
Example 6 As before, take your calculator and enter the number 0.3.
Second, program your machine to compute y = f (x) = 4(1- x) x.
Then, keep on doing the same as you did in the previous two examples.
Finally, analyse the output.
19
The plot of the sequence x
n
= 4(1- x
n-1
) x
n-1
(the first 50-ty terms)
Completely chaotic - divergent
20
LIMIT OF A SEQUENCE
21
If the limit of the sequence (a
n
) exists then we say the sequence (a
n
) is
convergent; otherwise, we say the sequence (a
n
) is divergent.
No matter how small the number is, at some ‘moment’ the terms of the sequence
enter the band of width and stay there
n
a
n
q
q+ε
q-ε
x
x
x
x
x
x
x
x
x
x x x
x
x
x
x
x
x
22
Suppose that the sequences (a
n
)
nN
and (b
n
)
nN
are convergent then for :
.
,
,
,
,
,
,
0
0
0
1
0
0
0
INDETERMINATE FORMS:
,
b
b
lim
,
a
a
lim
n
n
n
n
,
0
A
A
)
A
(
lim
.
6
R
C
,
a
)
a
(
lim
.
5
0
b
lim
if
b
a
b
a
lim
.
4
b
a
)
b
a
(
lim
.
3
,
b
a
)
b
a
(
lim
.
2
a
C
)
a
C
(
lim
.
1
a
a
n
C
C
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
R
C
23
How to use the definition to find the limit.
0
n
)
1
(
lim
n
n
24
1.
Guess
25
26
27
28
29
Theorem
If the limit exists, it must be unique.
If a
n
is convergent then it has only one limit.
Proof
Suppose that q’, q’’ are limits of (a
n
) and that
.Let N’ be a natural number such that if then
Let N” be a natural number such that if then
"
q
'
q
and
''
q
'
q
,'
N
n
3
1
'
q
a
n
,
"
N
n
3
1
"
q
a
n
Let N = sup{N’,N”}, so that
3
2
3
1
3
1
"
q
a
a
'
q
"
q
a
a
'
q
"
q
'
q
n
n
n
n
3
2
"
q
'
q
and
"
q
'
q
We obtain a contradiction
QED
n
n
a
lim
30
0
q
n
n
lim
1
q
0
q
n
q
n
log
log
1
q
n
0
log
log
IMPORTANT LIMITS
1.
,for
Sketch of proof:
so
QED
1
a
lim
n
n
1
a
2
n
x
2
1
n
n
nx
1
x
1
R
x
N
n
,
,for
For proof we need the Bernoulli Inequality
,
where
2.
QE
D
31
Theorem SQUEEZE PRINCIPLE
( Sandwich, Policeman Theorem )
n
n
n
c
,
b
,
a
n
n
n
c
b
a
q
c
lim
a
lim
n
n
n
n
q
b
n
n
lim
Assume that the sequences
satisfy
.
then
0
n
n
some
for
an
d
32
n
n
n
n
n
5
4
3
lim
n
n
n
n
n
n
n
n
5
3
5
4
3
5
EXAMPL
E
n
n
n
5
3
n
n
5
5
33
.
lim
1
n
n
n
34
Theorem
Every convergent sequence is bounded.
35
Note:
Boundedness is necessary but not sufficient to guaranty convergence.
Example:
(-1)
n
bounded,
divergent
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Theorem
BOUNDED MONOTONE SEQUENCE
THEOREM
Let a
n
be a monotone sequence
( increasing a
n
≤ a
n+1
or
decreasing a
n+1
≤ a
n
for all a
n
, except maybe some first terms a
1
, a
2
,
a
3
,... a
k
)
(i)
If a
n
is bounded
(appropriately:a) bounded above i.e. there
exists an upper bound U such that a
n
≤ U
or b)bounded below i.e.
there exists a lower bound L such that L
≤ a
n+1
)
then a
n
it is
convergent.
(ii)
If a
n
is unbounded then it is divergent
to either .
http://demonstrations.wolfram.com/ConvergenceOfAMonotonicSequence/
37
.
!
lim
0
n
a
n
n
38
IMPORTANT EXAMPLE THE NUMBER e:
The definition (existence) of number e – the base of the
natural logarithm,
the Euler number
n
n
n
1
1
x
We will use the monotone bounded sequence theorem to prove the existence of
n
n
n
1
1
lim
e
39
1.
We will show that
)
(
n
x
is an increasing sequence
2
1
1
2
1
1
2
1
2
n
3
2
1
2
1
2
1
2
n
1
3
1
2
1
2
x
1
n
2
n
!
!
!
2. And that (x
n
) is bounded
QED
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e ≈ 2,718281828459045....
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Definition
The sequence (a
n
) diverges to positive infinity iff
Example
The sequence is divergent to infinity
M
a
n
M
a
lim
n
n
n
Definition
The sequence (a
n
) diverges to minus infinity iff
N
n
n
2
)
(
M
a
n
M
a
lim
n
n
n
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Definition
A subsequence of the sequence (x
n
) is a sequence of the form
( x
a(n)
), where the a(n) are natural numbers with a(n) < a(n+1)
for all n.
Intuitively, a subsequence omits (loses) some elements of the
original sequence.
Theorem
A sequence is convergent if and only if all of its
subsequences converge towards the same limit.
43
.
,
,
,
,
,
,
0
0
0
1
0
0
0
INDETERMINATE FORMS:
44
EXTENDED ARITHMETIC
For the sake of convenience in dealing with indeterminate forms, we
define the following arithmetic operations with real numbers, positive
infinity and negative infinity.
Let c be a real number and c > 0, then we define:
)
)(
(
)
)(
(
)
)(
(
.
6
0
)
(
)
(
.
5
0
c
0
c
0
c
,
0
c
.
4
,
.
3
)
(
)
c
(
,
)
(
)
c
(
,
)
(
c
,
)
(
c
.
2
,
c
.
1
c
c
45
0
p
,
n
1
lim
0
p
,
0
n
1
lim
p
n
p
n
1
q
,
q
lim
1
q
,
0
q
lim
n
n
n
n
1
a
,
1
a
lim
n
n
1
n
lim
n
n
0
k
,
1
a
,
n
a
lim
k
n
n
0
!
n
a
lim
n
n
A
a
n
a
e
a
A
1
lim
n
n
;
;
,
;
;
;
.
IMPORTANT LIMITS
46
Let
n
n
b
a
( a
n
is „slower” then b
n
) if
0
b
a
lim
n
n
n
then for some n > n
0
:
n
n
n
n
n
k
3
2
a
n
n
!
n
a
2
n
n
n
n
n
log
47
Examples