CHAPTER 7
SEQUENCES AND SERIES
SEQUENCE：A sequence is a function where the domain is a set of consecutive positive integers beginning with 1. FINITE SEQUENCE：A finite sequence is a function having for its domain a set of positive integers，{1, 2, 3, 4, 5, …, n}，for some positive integer n. INFINITE SEQUENCE：An infinite sequence is a function having for its domain the set of positive integers, {1, 2, 3, 4, 5, …,n ,
}.
…
The function values are considered the terms of the sequence. The first term of the sequence is denoted with a subscript of 1, for example, a1 , and the general term has a subscript of n, for example, an . Example： Find the first four terms,
a10
and
a15
from the given nth term of the
sequence, an n 2 1, n 3.
Finding the General Term ：When only the first few terms of a sequence s equence are known, we can often make a prediction of what the general term is b y looking for a pattern. Example：Predict the general term of the sequence 1, 2, 4, 8, 16,…. Solution：These are powers of two with alternating signs, so the general term might be
() ) . Sums and Series Series:
Given the infinite sequence a1 , a2 , a3 , a4 ,
, an ,
, the sum of the terms
a1 a2 a3
an is called called an an infinite series. A partial partial sum is the sum of of the first n terms a1 a2 a3 an . A partial sum is also called a finite series or nth partial sum, and is denoted S n . Sigma Notation:
The Greek letter (sigma) can be used to denote a sum when the
general term of a sequence s equence is a formula. Example:
The sum of the first four terms of the sequence sequence 3, 5, 7, 7, 9, . . ., 2k 1 , . . . 4
can be named
2k 1. 1
Recursive Definitions: recursion formula.
A sequence sequence may be defined recursively recursively or by using a
Such a definition lists the first term, or the first few terms, and
then describes how to determine the remaining terms from the given terms. Example:
Find the first 5 terms of the sequence defined by
a1 5, an 1 2an 3,
for n 1.
ARITHMETIC SEQUENCES AND SERIES
Arithmetic Sequences: Sequences:
A sequence sequence is arithmetic arithmetic if there exists a number d , called the
common difference, such that an1 an d for any integer n 1. Definition
A sequence with general term an+1 = an + d is called an arithmetic sequence. a n = nth term t erm and d = common difference difference th
n Term of an Arithmetic Sequence:
The nth term
of an arithmetic sequence is
given by an a1 n 1 d , for any integer n 1.
th
Example:
Find the 14 term of the arithmetic sequence 4, 7, 10, 13, . . .
Example:
Which term is 301 from the sequence above?
Sum of the First n Terms of an Arithmetic Sequence Consider the the arithmetic sequence 3, 5, 7, 9, . . . the sequence, we get arithmetic series.
S 4 ,
which is 3 + 5 + 7 +
When we add the first four terms of 9, or 24.
This sum is called an
To find a formula for the sum of the first n terms,
S n ,
of an
arithmetic sequence, we first denote an arithmetic sequence, sequence, as follows:
a1 , a1
d , a1
2d ,
,
a
n
2d , a n
This term is 2 terms bac back from from the last
Sn a1 a1 d a1 2d
d
,a
n
.
This is the next to last ter term.
an 2d an d an .
reversing the order gives us
Sn an an d an 2d adding these two sums we have,
a1 2d a1 d a1.
2Sn a1 an a1 d an d a1 2d an 2 d
an 2d a1 2d an d a1 d an a1 .
Notice that all of the brackets simplify to a1 an and that a1 an is added n times. This gives us
2Sn n a1 an or Sn
n
2
a1 an
So the sum of the first n terms of an arithmetic sequence is given by
Sn
n
2
a1 an
GEOMETRIC SEQUENCES AND SERIES
A sequence sequence is geometric geometric if there is a number r, called
GEOMETRIC SEQUENCE: the common ratio, such that
an 1 an
r,
an 1 an r ,
or
for any intege integerr n 1.
Definition A sequence with general term an+1 = an r
is called an geometric sequence. a n = nth term and r = common ratio th
n TERM OF A GEOMETRIC SEQUENCE:
The nth term of a geometric is given
by
an a1 nr1 , SUM OF THE FIRST n TERMS:
The sum of the first n terms of a geometric
sequence is given by
Sn
n a1 1 r
1 r
,
f o r a n y i nn t
for any r 1.
INFINITE GEOMETRIC SERIES:
The sum of of the terms of an an infinite geometric
sequence sequence is an infinite geometric geometric series. series.
For some geometric geometric sequences,
close to a specific number as n gets very large.
S n
gets
For example, consider the infinite
series
1 2
1
1
1
4
8 16
1 2
n
LIMIT OR SUM OF AN INFINITE GEOMETRIC SERIES When r 1, the limit or sum of an infinite geometric series is given by
S
a1
1 r
. th
The rule used to generate a sequence, is often described by referring to the n term. A sequence can develop in 4 ways
Divergent
The terms keep growing The terms converge on a
Convergent
single value, in this case 0. The sequence repeats itself
Periodic
after a set number of terms. The sequence oscillates
Oscillating
between 2 values.
Algebra of sequences:
Given any two sequences { an} with limit value A, {bn} with limit value B, and any two scalars
k , p,
the following are always true:
(a){k an + p bn } is a convergent sequence with limit value
kA + pB
(b){ an bn } is a convergent sequence with limit value AB (c){ an / bn } is a convergent sequence with limit value A/B
provided that B 0
x) is a continuous function with lim f ( x ) L , and if an = f (n) for all (d)if f ( x x
values of n
then {an} converges and has the limit value L
(e)if an cn bn ,
then
{cn}
converges with limit value
C where A C B
Item (d) above permits us to use methods from the theory of functions, for example L’Hôpital’s rule, and in item (e) above above if the limit values of the sequences {bn}
are both the same, same, then this this is called the squeeze squeeze theorem.
{ an} and
If each element of a sequence
{ an}
( a1 a2
is no less than all of its predecessors predecessors
a3 a4 ...) then the sequence is called an increasing sequence. If each element of a sequence {an} is no greater than all of its predecessors (a1
a2 a3 a4 ...) then the sequence is called a decreasing sequence. sequence.
A monotonic sequence is one in which the elements are either increasing or decreasing.
an
If there exists a number M such that
M for all values of n
then the
sequence is said to be bounded. Theorem
Every bounded monotonic sequence is convergent. convergent.
Standard Sequences: Some of the most important sequences are (1) r n r (2) n
r 1 , r 2 , r 3 , .
1r , 2 r , 3r , .
1 < r 1. This sequence converges whenever r 0. This sequence converges whenever
th
General ( n n ) Term Test (also known as the Divergence Test):
If
lim a n 0 , then the series
n 1
n
an
diverges.
NOTE: This test is a test for divergence only, and says nothing about convergence. Geometric Series Test:
A geometric series has the form
n 0
a r n , where “a” is some fixed scalar (real
number). a
A series of this type will converge provided that r < 1, and the sum is
1 r
.
A proof of this result follows. Consider the
th
k partial sum, and “r ” times the
S k r S k The difference between Provided that
th
k
partial sum of the series
a ar 1 ar 2 ar 3 ar k ar 1 ar 2 ar 3 ar k ar k 1 rS k k and
S k k is
r 1, we can divide by
r 1S k
( r 1), to obtain
a r k 1 1 . k 1
S k
a r
r 1
.
1
Since the only place that “k ” appears on the right in this last equation is in the numerator, the limit of the sequence of partial sums { S k k} will exist iff the limit as
k
exists as a finite number. This is possible iff r < 1, and if this is true then the limit value of the sequence of partial sums, and hence the sum of the series, is S
a
1 r
.
Telescoping Series:
Generally, a telescoping series is a series in which the general term is a ratio of polynomials in powers of
“n”.
The method of partial fractions (learned when
studying techniques of integration) is normally used to rewrite the general term, and then the sequence of partial partial sums sums is studied.
This sequence sequence will, most most of the time,
simplify to just a few terms, and the limit can can then be determined.
One example of a
telescoping series will be presented here, and additional examples in class. As with sequences the main areas of interest with series are: (a) the determination of the general term of the series if the general term is not given, and (b) finding out whether or not the sum of the given series exists. Again as with sequences the determination of the general term of the series, if the general term is not given, relies heavily on pattern recognition. For a series that contains only finitely many terms, the sum always exists provided that each of the terms of the series is finite. For a series that contains infinitely many terms we need to use the following theorem. Theorem:
A series converges iff the associated sequence of partial sums
represented by {S k k} converges.
The element
S k k
in the sequence above is
defined as the sum of the first “ k ” terms of the series. In the remaining sections of this chapter, a number of different kinds of series will be considered. considered. They, generally speaking, fall into one of the following categories: (a) telescoping series (b) geometric series (c) hyperharmonic series (also known as p-series) (d) alternating series (e) power series (f)
binomial series
(g) Taylor series
We will also consider a number of tests that make it unnecessary to use the theorem mentioned above. (i)
The various tests that will be studied are:
th
n term test (also known as the divergence test)
(ii) geometric series test (iii) integral test (iv) comparison tests (v) alternating series test (vi) ratio test (vii) root test For a series with both positive and negative terms it is necessary to consider two different kinds of convergence. convergence.
These are conditional convergence, convergence, and absolute
convergence. If a series contains only positive terms, then conditional convergence is impossible, and we usually refer simply to convergence in this case. Properties of series:
(a)adding or deleting a finite number of finite terms in a given series has no effect on the convergence of the given series (b)if the series
an converges and has sum
converges and has sum B, and if p and q
A, and if the series
bn
are any finite constants, then
pan + q bn) converges and has sum ( pA + qB). ( pa
