Chapter 7 Sequences and Series (SAT course)

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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,

1a , and the

general term has a subscript of n, for example, .n

a

Example： Find the first four terms, 10a and 15

a from the given nth term of the

sequence, 2 1, 3.na n n

Finding the General Term：

When only the first few terms of a sequence are known,we can often make a prediction of what the general term is by 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 1 2 3 4, , , , , ,n

a a a a a , the sum of the terms

1 2 3 na a a a is called an infinite series. A partial sum is the sum of the

first n terms 1 2 3 .na a a a A partial sum is also called a finite series or nth

partial sum, and is denoted .nS

Sigma Notation: The Greek letter (sigma) can be used to denote a sum when the

general term of a sequence is a formula.

Example: The sum of the first four terms of the sequence 3, 5, 7, 9, . . ., 2 1k , . . .

can be named

4

1 2 1 .k



Recursive Definitions: A sequence may be defined recursively or by using a

recursion formula. 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

1 15, 2 3, for 1.n na a a n

ARITHMETIC SEQUENCES AND SERIES

Arithmetic Sequences: A sequence is arithmetic if there exists a number d , called the

common difference, such that 1n na a d for any integer 1.n

Definition

A sequence with general term an+1 = an + d is called an arithmetic sequence.

a n = nth term and d = common difference

nth

Term of an Arithmetic Sequence: The nth term of an arithmetic sequence is

given by 1 1 ,na a n d for any integer 1.n

Example: Find the 14th

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 arithmetic sequence 3, 5, 7, 9, . . . When we add the first four terms of

the sequence, we get 4S , which is 3 + 5 + 7 + 9, or 24. This sum is called an

arithmetic series. To find a formula for the sum of the first n terms,n

S , of an

arithmetic sequence, we first denote an arithmetic sequence, as follows:

1 1 1

This term is 2 terms This is the next

back from the last to last term.

, , 2 , , 2 , , .n n n

a a d a d a d a d a

1 1 1 2 2 .n n n nS a a d a d a d a d a

reversing the order gives us

1 1 12 2 .n n n nS a a d a d a d a d a

adding these two sums we have,



1 1 1

1 1 1

2 2 2

2 2 .

n n n n

n n n

S a a a d a d a d a d

a d a d a d a d a a

Notice that all of the brackets simplify to 1 na a and that 1 n

a a is added n times.

This gives us

1 12 or

2n n n n

nS n a a S a a

So the sum of the first n terms of an arithmetic sequence is given by

12

n n

nS a a

GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCE: A sequence is geometric if there is a number r, called

the common ratio, such that

11, or , for any integer 1.n

n n

n

ar a a r n

a

Definition

A sequence with general term

an+1 = an r

is called an geometric sequence. a n = nth term and r = common ratio n

thTERM OF A GEOMETRIC SEQUENCE: The nth term of a geometric is given

by

1

1 , f o r a n y i n tn

na a r n

SUM OF THE FIRST n TERMS: The sum of the first n terms of a geometric

sequence is given by

1 1, for any 1.

1

n

n

a r S r

r



INFINITE GEOMETRIC SERIES: The sum of the terms of an infinite geometric

sequence is an infinite geometric series. For some geometric sequences,n

S gets

close to a specific number as n gets very large. For example, consider the infinite

series

1 1 1 1 1

2 4 8 16 2n

LIMIT OR SUM OF AN INFINITE GEOMETRIC SERIES

When 1,r the limit or sum of an infinite geometric series is given by

1 .1

aS r

The rule used to generate a sequence, is often described by referring to the nth

term.

A sequence can develop in 4 ways

Divergent The terms keep growing Convergent The terms converge on a

single value, in this case 0. Periodic The sequence repeats itself

after a set number of terms. Oscillating The sequence oscillates

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

(d)if f ( x) is a continuous function with L x f x

)(lim , and if an = f (n) for all

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 if the limit values of the sequences {an} and{bn} are both the same, then this is called the squeeze theorem.



If each element of a sequence {an} is no less than all of its predecessors (a1 a2

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.

A monotonic sequence is one in which the elements are either increasing or decreasing.

If there exists a number M such that an M for all values of n then the

sequence is said to be bounded.

Theorem Every bounded monotonic sequence is convergent.

Standard Sequences:

Some of the most important sequences are

(1) ,,, 321 r r r r n . This sequence converges whenever 1 < r 1.

(2) ,3,2,1 r r r r n . This sequence converges whenever r 0.

General ( nth

) Term Test (also known as the Divergence Test):

If 0lim

nn

a , then the series

1n na diverges.

NOTE: This test is a test for divergence only, and says nothing about convergence.

Geometric Series Test:

A geometric series has the form

0n

nr a , where “a” is some fixed scalar (real

number).

A series of this type will converge provided that r < 1, and the sum isr

a

1.

A proof of this result follows.

Consider the k th

partial sum, and “r ” times the k th

partial sum of the series

1321

321

k k

k

k

k

ar ar ar ar ar S r

ar ar ar ar aS

The difference between rS k and S k is 11 1 k

k r aS r .

Provided that r 1, we can divide by (r 1), to obtain

1

11

r

r aS

k

k .



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 } 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

r

aS

1.

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 sums is studied. This sequence will, most of the time,

simplify to just a few terms, and the limit 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 thateach 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 } converges. The element S 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. 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. The various tests that will be studied are:

(i) nth

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. These are conditional 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 A, and if the series bn

converges and has sum B, and if p and q are any finite constants, then

( pan + q bn) converges and has sum ( pA + qB).

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Chapter 7 Sequences and Series (SAT course)