Limit of a Sequence

Limit of a Sequence

Advanced Level Pure Mathematics

Advanced Level Pure Mathematics

Calculus I

Chapter 7Limit of a Sequence

7.1Introduction2

7.2Sequences2

7.3Convergent Sequences5

7.4Divergent Sequences and Oscillating Sequences6

7.5Operations on Limits of Sequences7

7.6Sandwich Theorem for Sequences13

7.7Monotonic Sequences15

7.8The Number e21

7.9Some Worked Examples22

7.1Introduction

Some examples of sequences:

1.1, , , ,  , , 

2.1, 1, 1, 1,  , (1)r+1 , 

3.cosx , cos2x , cos3x ,  , cosrx , 

7.2 Sequences

Definition 7.1A sequence {xn} is a function on the set of real numbers, and is usually written as

x1 , x2 , x3 ,  , xn , .

1.The term xn is called the general term of the sequence.

Example{1, 2, 4, 8, ...} is a sequence of positive integers with general term .

2.If the sequence has infinite number of terms, it is called an infinite sequence.

3.If the sequence has finite number of terms, it is called a finite sequence.

4.Sn = x1 + x2 + x3 +  + xn +  is said to form a series.

5.Sn == x1 + x2 + x3 +  + xn is a finite series.

6.Sn == x1 + x2 + x3 +  + xn +  is an infinite series.

How to find the series sum , ? By using

(M1)Mathematical induction.

(M2)Method of difference where such that

(M3)Partial fractions and method of difference.

(M4)Standard formulae:

(i)A.P. : a , a+d , a+2d ,  , a+(n1)d ,  .

Sn = .

(ii)G.P. : a , ar , ar2 ,  , arn1 ,  .

Sn = ; S = where |r| < 1 .

(iii)1 + 2 + 3 +  + n = =

12 + 22 + 32 +  + n2 = =

13 + 23 + 33 +  + n3 = =

(M5)Arithmetic-Geometric Series (A.G.P.)

Sequence : a , (a+d)R , (a+2d)R2 ,  , [a+(n1)d)Rn1 ,  .

Sn =.

In particular, 1 , 2x , 3x2 ,  , nxn1 ,  .

Sn =.

(M6)Harmonic Series

Hn=

There is no simple formula for the sum of the first n terms.

(M7)Difference equation / Recurrence relation

Recurrent sequence :

(1)a n+1 = Aanfor n = 1 , 2 , 3 , ... .

(2)an+2 = Aan+1 + Ban for n = 1 , 2 , 3 , ... .

where the coefficients A and B are constants.

For case (1), we have

an+1 = Aan = A(Aan1) = A2 an1 = A3 an2 == Ar a1

If a1 = Aa , then an+1 = An+1a.

For case (2) , we have

an+2Aan+1Ban = 0 .

Let  ,  be the roots of the auxiliary equation t2 AtB = 0 .

(i)If , then an = c1n + c2n.

(ii) If  = , then an = (nc1 + c2 )n.

Example 1A sequence of real numbers an is defined by

a0 = 0 , a1 = 1 and an+2 = an+1 + an for all n = 0, 1, ... .

Show that for all non-negative integers n , an = (nn ), where  ,  are roots of the equation x2 + x 1 = 0 with .

Soln.an+2 = an+1 + an an+2 + an+1an = 0

Consider the auxiliary equation t2 + t 1 = 0 , we have t = .

 = and  =

Let an = c1 ()n + c2 ()n , where c1 , c2R.

Since a0 = 0 , we have

0 = c1 + c2 ...... (1)

Since a1 = 1 , we have

1 = c1 () + c2 () ...... (2)

On solving (1) and (2) , we obtain c1 = and c2 = .

Hence, an = (nn ) , where  ,  are roots of the equation x2 + x 1 = 0

with .

7.3Convergent Sequences

Definition 7.2A convergent Sequence {xn} is a sequence whose terms will approach a finite value a as n tends to infinite.

We say that xn a as n.

Symbolically, , which is called the limit of the convergent sequence.

ExampleIf xn = , the sequence {xn} is convergent to 0 .

ExampleThe sequence {} is convergent to 1.

ExampleThe sequence {} is also convergent to 0.

Definition 7.3A sequence {xn} is said to converge to a if and only if for any  > 0 , there exists a positive integer N such that when nN , we have |xn a| <  .

a is called the limit value of {xn} and we write .

Definition 7.4A sequence {xn} not convergent is called divergent.

Theorem 7.1The limit value of a convergent sequence is unique.

i.e.if xn a and xn b as n, then a = b.

Theorem 7.2All convergent sequences are bounded. i.e. | xn | Mfor all nN .

Important Facts:

(1)If , then .

(2)If , then for p = 1, 2, 3, ...

(3){xn } converges to a iff every subsequences of {xn} converges to a.

i.e.odd sequence x1 , x3 , x5 ,  and even sequence x2 , x4 , x6 , 

both converges to a.

(4) if | x | < 1.

(5).

(6) where x 0 .

(7).

(8)

7.4Divergent Sequences AND OSCILLATING SEQUENCES

Definition7.5A sequence {xn} is said to diverge to positive infinity if for any positive real number M, there exists a positive integer N such that when n N, xn > M.

We write .

A sequence {xn} is said to diverge to negative infinity if for any positive real number M , there exists a positive integer N such that when nN , xn < M.

We write .

Theorem 7.3Let {xn} be a sequence with xn 0 .

Then if and only if .

N.B.

Definition 7.6Oscillating sequences are neither convergent nor diverging to infinity.

Example{1, 1, 1, 1, 1, 1, ...} is an oscillating sequence.

Example{0 , 4 , 0 , 8 , ... , n[1+(1)n ] , ...} is a infinitely oscillating sequence.

Example(1)xn = sinn

(2)xn = ncosn

(3)xn = n (1)n

(4)xn = (1)n

7.5Operations on Limits of Sequences

Theorem 7.4Let {xn} and {yn} be two convergent sequences.

(a).

(b).

(c), where .

(d), where k is a constant.

(e)For any positive integer m,

(i),

(ii),

(iii).

Example 2Find (a) (b) .

a) HF p.60 (2.4a)

b) HF p.61 (2.5)

Example 3Find.

HF p.61 (2.6b)

*N.B.We cannot use these rules of operations if {xn} or {yn} is not convergent.

Example(, i.e. {n} diverges )

Example 4Find .

Wrong proof :

Since = 0 for kn.

So= 0 + 0 +  = 0

It is invalid since the sum of an infinite number of terms, each term tends to

zero, may not be zero.

HF p.62 (2.7)

Example 5Show that the limit value does not exist.

Soln.Whennis even, let n = 2k , where k is an integer. Then

=

=

= =

 =

HF p.63 (2.8)

Theorem 7.5Let {xn} and {yn} be two sequences.

(a) If and |yn| M for all n , then .

(b)If and |yn| M for all n , then .

(c)If and yn 0 for all nN and yn does not converge to 0 , then

Example 6Evaluate .

Soln.Since |sinn|  1 for all n and = 0 ,  = 0.

Example 7 Evaluate (a) (b) (c)

N.B.For Theorem 2.5 , yn does not necessarily exist.

Example 8 Evaluate .

HF p.67 (2.12)

ExampleA sequence is defined by

, and ()

Show that for all positive integers ,

Hence evaluate .

Solution

HF p.63 (2.9)

Example 9A sequence is defined by

and ()

Find .

Solution

HF p.65 (2.10)

7.6Sandwich Theorem for Sequences

Theorem If xnyn , then .

Theorem 7.6Sandwich Theorem for Sequences

Let {xn} , {yn} and {zn} be three sequences such that

xn  yn  zn

and

Then.

Example10Prove that .

Soln.Since1  sinn 1

So for n 1

Since

Hence, by sandwich theorem for sequence, .

Example 11Find .

since  for all k = 0 , 1, 2, . . ., n

HF p.72 (2.14)

N.B.It is wrong to say

=

= 0 + 0 + ...(totally infinitely many 0)

= 0

because is an indeterminate form.

Other indeterminate forms :

Example 12 Evaluate .

HF p.75 EX 2B (1c)

Example 13Let a be a real number greater than 1. Prove that .

Since a > 1, let a = 1+h where h is a positive real number.

Then an= (1+h)n =

HF p.72 (2.5)

Example 14Let A be a positive real number and {an} be a sequence of real numbers such that a1A and an+1 = for n 1.

(a)Show that anA for all positive integers n.

Hence, show that anA(an1A).

(b)Find by using sandwich theroem .

HF p.74 (2.18)

Theorem 2.7(a)If and there exists a positive integer N such that xnyn as nN ,

then .

(b)If and there exists a positive integer N such that xnyn as nN , then .

Example 15 Find . [compare with Example14 and 15]

Since
HF p.75 EX 2B (1d)

7.7Monotonic Sequences

Definition 7.7(1)A sequence {xn} is said to be monotonic increasing if and only if xnxn+1

for n = 1, 2, 3, ... .

(2)A sequence {xn} is said to be monotonic decreasing if and only if xnxn+1 for n = 1, 2, 3, ... .

(3)A sequence {xn} is said to be monotonic if and only if it is either increasing or decreasing.

(4)A sequence {xn} is said to be strictly increasing or strictly decreasing if and only if xnxn+1 or xnxn+1 for all n.

Example 16Show that the sequence {} is strictly decreasing.

Soln.Since for all nN

So the sequence {} is strictly decreasing.

Example 17Let a sequence {xn} be defined by xn = . Show that the sequence is

monotonic increasing.

HF p.77 (2.19b)

Definition 7.8(a)A sequence {xn} is said to be bounded above if and only if

there exists a constant M such that xnM for n = 1, 2, 3, ... .

(b)A sequence {xn} is said to be bounded below if and only if

there exists a constant M such that xnM for n = 1, 2, 3, ... .

Theorem 7.8(a)If a monotonic increasing sequence is bounded above, then

the sequence is convergent and limit of {xn} exists.

(b)If a monotonic decreasing sequence is bounded below, then

the sequence is convergent and limit of {xn} exists.

decreasing and bounded belowincreasing and bounded above

Example 18 Let the sequence {xn} be defined by xn = for n 1 .

(a)Show that {xn} is an increasing sequence and bounded above.

(b)Hence, show that {xn} is convergent.

HF p.78 (2.20)

N.B.(1){xn} converges and xnL , then L.

(2){xn} converges and xnL , then L.

Example 19 Let a and b be two positive real numbers. A sequence {xn} is defined by

for n 1. It is given that 0 < x1b.

(a)Show that xnb for all positive integer n.

(b)Show that {xn} is monotonic increasing. Hence, show that {xn} is convergent.

(c)Find .

HF p.82 (2.24)

Example 20Let a and b be two real numbers such that ab > 0 . Two sequences {an} and {bn} are defined by

, for n > 1 and , .

(a)Prove that {an} is monotonic decreasing. Hence deduce that {bn} is monotonic increasing.

(b)Prove that {an} and {bn} converge to the same limit.

HF p.83 (2.25)

Example21Let x1x2 > 0 and .

(a)Show that {x2n1} is a decreasing sequence and {x2n} is an increasing sequence .

(b)Show that x2n1x2m for all positive integers n and m .

(c)Hence, show that {x2n1} and {x2n} have a common limit and find it .

HF p.86 EX 2C (14)

Example 22Given two positive numbers a and b where ab and {an}, {bn} are two sequences defined by

, and , for all n 2,

(a)Show that bnbn+1an+1an .

(b)Show that {an} and {bn} have the same limit. Find this common limit.

BR p.326 (8.4)

7.8The number e

Consider the sequence of numbers defined by:

The following table give the value of the sequence corresponding to different values of n.

As the value of n increases without bounds, the value of increases steadily, but it seems to increase slower and slower. We can see that it would stop somewhere around 2.7182….

This number plays an important role in advanced mathematics and is denoted by e.

Definitione = , where n takes positive integral values.

Or

N.B.e is an important irrational number in calculus.

Theorem 7.92 e 3 .

Example 23 Find .

Soln.= = =

Example 24 Express the following limits in terms of e.

(a)(b) (c)

(d)

HF p.88 EX2D (2.26, 1b, c, f)

7.9SOME WORKED EXAMPLES

Example 35Let a1 = 2 , b1 = and an = an1 , bn = bn1 for n 2 .

(a)Prove that anbn and an bn = 2n + 1 for n 1 .

(b)Using (a), or otherwise, show that an2 > 2n + 1 for n 1 .

Hence find .[HKAL98](7 marks)

Example 26(a)Let x > 1 and define a sequence {an} by

a1 = x and an = for n 2 .

(i)Show that an > 1 and an > an+1 for all n .

(ii)Show that = 1 .(8 marks)

(b)Let f : [1,]  R be a continuous function satisfying

f(x) = f for all x 1 .

Using (a), show that f(x) = f(1) for all x 1.[HKAL96](7 marks)

Example 27 For any  > 0 , define a sequence of real numbers as follows :

a1 =  + 1 , an = an1 + for n > 1 .

(a)Prove that

(i)an2an12 + 2 for n 2 ;

(ii)an22 + 2n + 1 for n 1 .(2 marks)

(b)Using (a), show that for n 2 ,

an22 + 2n + 1 + .(3 marks)

(c)Prove that for k 1 ,

.(2 marks)

(d)Using the above results, show that exists and find the limit.

State with reasons whether exists.[HKAL95](8 marks)

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