5. Motivation Convergence and divergence in applied
mathematics
Convergence is the concept of an (infinite) process getting
closer and closer to some limiting state.
Divergence is where the process does not converge.
This is important to understand as it is fundamental to much of
applied mathematics, e.g.
Mathematical models of the real world are almost always only
approximations and so only offer approximations to the truth.
This is ok as long as we can in principle make the
approximations as good as we want, i.e. make them converge
to the actual truth.
6. Increasing detail of the IPCC reports 1990 - 2007
Source:
A National Strategy for Advancing Climate Modeling, US National
Academy of Sciences
7. Motivation Convergence and divergence in applied
mathematics
In addition . . .
The mathematical problems set up on these already
approximate models are almost always impossible to solve
exactly . . .
. . . and we can only find approximate solutions to them (so
called numerical solutions).
This is ok as long as we can in principle make the
approximations as good as we want, i.e. make them converge
to the actual solution.
For time based models, such as weather/climate, these
approximations get poorer and poorer as we look into the
future, i.e. they diverge from the actual solution.
8. Motivation Convergence and divergence in pure
mathematics
The integers Z and rationals Q can be rigorously described
without too much trouble,
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . } positive and negative whole num
Q =
n
m
: n, m ∈ Z, m = 0 ratios of integers.
(See Saeed’s Set Theory lectures for explanation of this
{ }notation.)
Non-rational numbers (so called irrationals) are harder to
describe and construct rigorously and precisely, e.g.
√
2, π, e
9. Motivation Convergence and divergence in pure
mathematics
One way to do so is to describe them as the limits of infinite
sequences or infinite sums of rational numbers, e.g.
π
4
=
∞
n=1
(−1)n+1
2n − 1
= 1 −
1
3
+
1
5
−
1
7
+ . . .
Many such interesting summation formulas exist for π.
Infinite sequences and sums of mathematical objects and their
convergence / divergence properties are a fundamental part of
calculus and analysis, i.e. the study of the number line (2-d
plane, 3-d space, . . . ) and functions defined on them and how
these functions change.
10. Motivation Convergence and divergence in pure &
applied mathematics
So there is lots of convergence & divergence going on
This needs to be studied and understood
We will study sequences and series and their
convergence/divergence.
11. Sequences Definitions and notation
Definition (1.1) A sequence is a list of elements (usually numbers)
indexed by the positive integers. We usually use a single letter, with
subscript, to denote the elements of a sequence, as in
x1, x2, x3, x4, . . . .
We can represent the sequence x1, x2, x3, x4, . . . using the compact
notation {xn}∞
n=1 or just simply {xn} if the n indexing is understood.
12. Sequences Examples
Example (1.1)
The sequence of positive integers 1, 2, 3, . . . , or {zm}, where
the rule is
zm = m.
The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... or {fn} , where
the rule is
f1 = 1, f2 = 1 and fm = fm−1 + fm−2 for all m ≥ 2,
i.e. each element is the sum of the previous two elements. This
type of rule for a sequence, where each term is defined using
the values of some previous terms, is called a recurrence
relation.
The sequence of odd positive integers: 1, 3, 5, 7, ...or {an},
where
an = 2n − 1.
13. Sequences Examples
The sequence of prime numbers: 2, 3, 5, 7, 11, 13, ... or {pm} ,
where
pm =???.
14. Sequences Arithmetic and geometric sequences
Definition (1.2) An arithmetic sequence has a common difference
between successive elements and a geometric sequence has a
common ratio of successive elements. So we can write an arithmetic
sequence as
a, a + d, a + 2d, a + 3d, . . .
or {an} , where an = a + (n − 1)d . Here a is the initial element and
d is the common difference. A geometric sequence can be written as
a, ar, ar2
, ar3
, . . .
or {an} , where an = arn−1. Here a is the initial element and r is
the common ratio.
15. Sequences Arithmetic and geometric examples
Examples (1.2 through 1.5)
The sequences of positive integers and odd positive integers
above are actually arithmetic sequences. The sequence of
positive integers can be written as {an} , where
an = n = 1 + (n − 1),
so 1 is the initial element and 1 is the common difference. The
sequence of odd positive integers can be written as {an} ,
where
an = 1 + (n − 1)2,
so 1 is the initial element and 2 is the common difference.
16. Sequences Arithmetic and geometric examples
The sequence,
1,
1
2
,
1
4
,
1
8
, . . .
of powers of 1
2, is geometric as it has the form {an} , where
an =
1
2
n−1
,
so 1 is the initial element and 2 the common ratio.
Compound interest awarded on investments or charged on
debts provide examples of geometric sequences. Suppose a
bank offers an annual compound interest rate of r% on an
initial deposit of P units. Then the value of the deposit after n
years is given by
vn = P 1 +
r
100
n
.
17. Sequences Arithmetic and geometric examples
The sequence {xn}, where
xn =
1
n2
,
is neither arithmetic nor geometric, as it is not possible to
express it in the required form.
18. Sequences Monotonic sequences: increasing or
decreasing behaviour
Definition (1.3) A sequence {xn} is increasing if its elements satisfy
x1 ≤ x2 ≤ x3 ≤ . . .
and decreasing if they satisfy
x1 ≥ x2 ≥ x3 ≥ . . . .
If the inequalities are all strict then the sequence can be called
strictly increasing or strictly decreasing as appropriate.
Proving monotonicity, i.e. increasing or decreasing behaviour
Two approaches . . .
Examine the difference between consecutive terms, i.e.
an+1 − an and then try to establish one of the inequalities
an+1 − an ≥ 0 or an+1 − an ≤ 0.
19. Proving monotonicity, i.e. increasing or decreasing
behaviour
Examine the quotient of consecutive terms,
an+1
an
,
and try to establish one of the inequalties
an+1
an
≥ 1 or
an+1
an
≤ 1.
Strict inequalities here will establish the strict version of
increasing / decreasing as appropriate.
20. Proving monotonicity, i.e. increasing or decreasing
behaviour
Example (1.6) Investigate the sequence {an} defined by
an =
5n
n!
,
and describe its increasing / decreasing behaviour.
NB: n! is the factorial of n, defined by,
n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1.
Investigate by computing some values. . .
. . . and then prove it.
21. Proving monotonicity, i.e. increasing or decreasing
behaviour
More examples
Try the difference or ratio method (or both) on the following
sequences to determine their increasing/decreasing nature
{sn} where sn = 3n
4n+1,
{fn} where fn = n2
2n ,
{yn} where yn = 1√
n2+1−n
.
Confirm this behaviour and investigate their convergence by
computing elements from the sequence.
