Transcript Mathematics
Mathematics
Medicine
Sequences and series
2010-2011
Sequences
Definition A sequence is an ordered set of numbers
Each number in the sequence is called a term of the sequence.
The term corresponding to the positive integer n is the n-th term of the sequence
and is denoted by a symbol such as bn.
The sequence whose n-th term is bn is denoted {bn}.
If sequences have a finite number of terms, they are called finite sequences.
Some sequences go on for ever, and these are called infinite sequences.
Sequences
There are two common ways used to describe a sequence:
1. Stating the general term. For example:
bn 2n 3 n 1,2,3,...
2. Giving the first term, b1, and stating the relationship between each
term and its successor. For example:
b1 1, bn1 bn 2n
n 1,2,3,...
When the second procedure is used to define a sequence, the sequence
is said to be defined recursively or by means of a recursion formula.
Arithmetic sequence
Definition An arithmetic sequence is a sequence with general term:
an a1 d n 1
where a1 is the first term and d is a constant (common difference).
Geometric sequence
Definition A geometric sequence is a sequence with general term:
an a1r
n1
where a1 is the first term and r is a constant (common ratio).
Convergent and divergent sequence.
It can happen that as we move along the sequence the terms get closer
and closer to a fixed value A (called limit).
lim an A
n
The notation is read:
“The sequence {an} converges to A” or “The limit of {an} is A”.
If a sequence has a limit, it is a convergent sequence. If a sequence
has no limit, it is a divergent sequence.
The Limit of a Sequence
EXAMPLE 1. The terms of sequence 1 are very small for large values of n. The terms are clustered
very close to zero for large n, but zero is not a term of the sequence. Zero is the limit of the sequence.
The sequence is said to converge to zero.
EXAMPLE 2. Each term in sequence 2 is three greater than its predecessor. For large values of n, the
terms are large. There is no number about which the terms of the sequence cluster. This sequence has no
limit; it is said to diverge.
EXAMPLE 3. Sequence 3 is similar to sequence 1. The terms are close to zero for large values of n, but
unlike sequence 1, there are terms to the right and to the left of zero. This sequence converges to zero.
EXAMPLE 4. The terms of sequence 4 are very close to 3 for large values of n. Since the terms are
close to 3, the sequence has a limit of 3.
EXAMPLE 5. The terms of sequence 5 which correspond to even numbers are close to 2. The terms
corresponding to odd numbers are close to -2. Consequently there is not a unique single number about
which the terms cluster. The sequence has no limit. Therefore it is to diverge.
The discussion of the examples suggests the
following ideas:
a. Some sequences have limits; some do not.
b. If a sequence has a limit, then the limit is a unique number.
c. If a sequence has a limit, then most of the terms are close to the limit.
Series and Their Sums
Definition A series is an indicated sum of the terms of a sequence.
EXAMPLE 1. Given the arithmetic sequence a, a + d, a + 2d,…, a + (n-1)d
the related arithmetic series is
a + (a + d) + (a + 2d) + … + [a + (n -1) d
2
3
n-1
EXAMPLE 2. Given the geometric sequence a, ar, ar , ar , …, ar
the related geometric series is
a + ar + ar2 + ar3 +… +arn-1
Partial sum
Definition Given a series . The nth partial sum, Sn, is defined by the equation
Sn a1 a2 ... an ...
Definition Given an infinite series
a1 a2 ... an ...
The sum of the infinite series is defined to be the limit of the sequence
of partial sums {Sn} where
Si a1 a2 ... ai
for all i; in the set of natural numbers.
If {Sn} has a limit, the series is a convergent series.
If {Sn} has no limit, the series is a divergent series.
Arithmetic series
If the terms of an arithmetic sequence are added, the result
is known as an arithmetic series.
The sum of the first n terms of an arithmetic series with first term a and
common difference d is denoted by Sn and given by
n
Sn (2a (n 1)d)
2
Geometrics series
If the terms of a geometric sequence are added, the result
is known as a geometric series.
The sum of the first n terms of a geometric series with first term a and common
ratio r is denoted by Sn and given by
a(1 r )
Sn
1r
n
provided r is not equal to 1
The formula excludes the use of r = 1 because in this case the
denominator becomes zero, and division by zero is never allowed
Sum of an infinite sequence
When the terms of an infinite sequence are added we obtain an infinite series.
But in in some cases sum is finite and can be found.
Consider the special case of an infinite geometric series for which the common
ratio r lies between -1 and 1.
In such a case the sum always exists, and its value can be found from the
following formula:
The sum of an infinite number of terms of a geometric series is denoted by S∞
and is given by
provided -1 <r< 1
Note that if the common ratio is bigger than 1 or less than -1,
that is r>1 or r<-1, then the sum of an infinite geometric series cannot be found.
Limits of Special Sequences
Theorem Any sequence
For example
bn
1
1 0
n 2
1
na
with a > 0 and a R converges to zero
Theorem Any sequence {c} of constants c converges to c.
For example: {5} 5
Theorem If | r | < 1, then {rn} converges to 0.
9 n
For example:
0
10
Operations with Sequences
Definitions Given any two sequences {an} and {bn}, then
•
the sum of {an} and {bn} is the sequence with nth term an + bn;
–
•
the difference of {an} and {bn} is the sequence with nth term an - bn;
–
•
that is {an}+{bn}={an+bn}.
that is {an}-{bn}={an-bn}.
the product of {an} and {bn} is the sequence with nth term an bn;
–
that is {an}{bn}={anbn}.
an
• the quotient of {an} and {bn} is the sequence with nth term bn
an
bn
that is {an}{bn}=
Theorem If {an} A and {bn}B, then
a.{an+bn}A+B.
b. {an-bn}A-B.
c. {anbn}AB.
d. (B0)
an
A
B
bn