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1
Introduction to
Microeconomics
Lecture 7
Maths: Sequences, Series and Limits;
the Economics of Savings & Investment
Ian Crawford
Outline
z
Sequences and series
z
Arithmetic and geometric
z
Deposits and interest
z
Present Value
Present Value
z
Savings and borrowing decisions
z
Limits
z
The number
e
and some applications
Sequences: order set of
numbers
z
A
SEQUENCE
is a set of numbers arranged in a definite
order, e.g. (i)
2, 4, 6, 8,…; (ii) 1, 4, 9, 16, 25,…
z
The dots indicate that these are infinite sequences.
Sequences may be finite.
z
There may be a formula that defines the terms e.g. for
sequence (ii) the
n
th
term,
u
n
, is
u
n
= n
2
z
A formula may use the preceding terms: “a recurrence
relation”.
z
An example is
u
1
= 1,
u
2
= 1,
u
n
= u
n
1
+
u
n
2
z
This gives the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,…
Series: sum of a sequence
z
A
SERIES
is formed when the terms of a sequence are
added together.
z
Notation:
12
...
n
rn
uuu
u
≡
+++
∑
Notation:
z
Here the notation
Σ
(pronounced “sigma”) denotes “the
sum of”
1
r
=
Series: sum of a sequence
z
Some times it’s handy to express a product as a sum.
z
Notation:
...
≡×××
∏
n
u
z
Here the notation
∏
denotes “the product of”
z
By taking logs it becomes a sum:
1
=
r
1
1
ln
ln
=
=
⎛⎞
≡
⎜⎟
⎝⎠
∑
∏
n
n
rr
r
r
uu
Arithmetic and Geometric
Sequences
z
An ARITHMETIC SEQUENCE is one in which
the
difference
between successive terms is
constant, such as 2, 4, 6, 8,…
z
Here the
common difference
is 2
z
A GEOMETRIC SEQUENCE is one in which the
ratio
of successive terms is a constant such as ½,
1, 2, 4, 8.
.
z
Here the
common ratio
is 2
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Geometric series
z
What is the sum of the first
n
terms in a geometric
sequence? Assume the common ratio
r
≠
1
S
n
=
a
+
ar + ar
2
+
ar
3
+ … +
ar
n
1
= a
(1 –
r
n
)/(1 –
r
)
z
Proof (Workbook, Chapter 3, Sect. 2.5)
S
n
=
a
+
ar + ar
2
+
ar
3
+…+
ar
n
1
rS
n
=
ar
+
ar
2
+
ar
3
+ … +
ar
n
1
+
ar
n
S
n
–
rS
n
= a – ar
n
which then gives the formula
Application: finance
z
You have £500 in the bank and the interest rate is 6% (i.e. 0.06) per
year. Interest is paid at the end of the year.
z
At the end of year 1 you receive 0.06 x 500 = £30
z
The interest is added to the £500, so you have £530
z
At the end of year 2 you receive 0.06 x £530 = £31.80, so your total
sum is £561.80
z
Generally: if you invest
P
and interest is paid annually at interest rate
i
then after one year you have
P
(1 +
i
)
z
After two years you have
P
(1 +
i
)(1 +
i
) =
P
(1 +
i
)
2
z
After
t
years you have
P
(1 +
i
)
t
z
The amount of money in your bank at the end of each year is a
geometric sequence with common ratio (1 +
i
)
How much will pension
contributions be worth?
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 Spring '10
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 Economics, Microeconomics

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