Lecture - 2

1

Convergence of Sequences

Contents

• Principle of Convergence• Algebra on limits• Divergence• Cauchy Sequence

2

Convergent SequencesConsider the sequence (1/n) and few of its terms

3

n an

1 1

2 1/2 = 0.5

10 1/10 = 0.1

100 1/100 = 0.01

1000 1/1000 = 0.001

10000 1/10000 = 0.0001

100000 1/100000 = 0.00001 etc

Convergent Sequences – Definition

• A sequence (an) is said to converge to a number l if given

> 0, there exists a positive integer m such that |an - l| <

for all n ≥ m.

• We say that l is the limit of the sequence (an) 4

Convergent Sequences – Definition

|an - l| < => - < an – l <

=> l - < an < l +

We write or (an) l as n ∞

if and only if

given > 0, there exists a positive integer m such that an (l-, l+) for all n ≥ m.

5

nn

alim

Convergent SequencesRevisit the sequence (1/n) and few of its terms

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m |an – 0| = |an| n ≥ m

0.1 11 0.09,0.08,0.07….. < 0.01 101 0.0099, 0.0098…. < 0.12 9 0.1111,0.1…. <

0.003 334 0.00299…. < 1 2 0.5……<

1.5 1 1,0.5,….. <

Convergent Sequences – Example Cont’d

A Constant Sequence 2, 2, 2……

For any n, an = 2,

So we guess that limit of this sequence could be = 2.

For |an-2| = |2 – 2| = 0 <

= 27

nn

alim

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Convergence of Sequences

n allfor 2

odd isn if 11)(

even isn if 1)(1

11aa 1nn1nn

Consider ((-1)n). As n varies sequence will have 1 or -1. It may not be easy to guess a “limit”. First have a closer look:

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Sequence – ((-1)n) Cont’d

arbitrary is 0ε because

not true is which ε,1or 2ε2 isThat

2

11

1111,Also

1nn

1nn1nn

ll

ll

1. If a > 0 is any real number, then

(a1/n) 1

2. (n1/n) 1

3.

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Some Sequences

0n

nsin

The sequences

and (1+1/n)n

converge a number 2 < e < 3

The geometric sequence (rn) converges if -1 < r ≤ 1

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Some Sequences

n!

1........

3!

1

2!

1

1!

11a n

Two sequences (an) a, and (bn) b, then

i. (an + bn) a + b

ii.(an - bn) a – b

iii.If k R, then (kan) ka

iv.(anbn) ab

v.(an/bn) a/b bn 0 for all n; b 012

Algebra of Limits

vi. If an ≤ bn for all n then a ≤ b

vii. If an ≥ 0 for all n and a 0,then

viii. If an ≤ cn ≤ bn for all n and a = b then (cn) a

ix. If (an) 0 and (bn) is bounded then (anbn) 0

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Algebra of Limits

aa n

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Example - 1

3003 n

5lim

n

2lim3

n

5

n

23limNumerator Now

)7/n4/n(6n

)5/n2/n(3nlim

74n6n

52n3nlim

2nn

2n

22

22

n2

2

n

15

Example – 1 Cont’d

2

1

6

3

74n6n

52n3nlimHence,

6r denominatoSimilarly

2

2

n

16

Example – 2

3

1

6

2

n

12

n

11

6

1lim

6n

)12)(1(lim

n

......21lim

n

3n3

222

n

nnnn

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Example – 2

0n

1-lim ) b(alim Hence,

bounded is 1- )(b

and 0 n

1)(a .

n

1-lim

n

nnn

n

nn

n

n

n

A sequence (an) is said to diverge to ∞ if given any real number k > 0, there exists no N such that an > k for all n ≥ no

We write (an) → ∞

A sequence (an) is said to diverge to -∞ if given any real number k < 0, there exists

no N such that an < k for all n ≥ no

We write (an) → - ∞

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Divergent Sequences

• A sequence (an) which is neither convergent nor divergent to ∞ or - ∞ is said to be an Oscillating Sequence.

• Further if it is bounded, then is said to be finitely oscillating;

• If not bounded it is said to be infinitely oscillating

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Oscillating Sequences

• (n) = 1, 2, 3, ………

• (n2) = 1, 4, 9,………….

• (-n) = -1, -2, -3, ………….

• (rn) diverges if r > 1

• ((-1)n) = -1, 1, -1…………….

• Z = 0, 1, -1, 2, -2,…………………

• (rn) oscillates if r ≤ -1

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Sequences - Examples

A sequence (an) is said to be a Cauchy Sequence if given > 0, there exists no N such that |an – am| < for all n, m ≥ no

Equivalently, |an+p – an| < for all n ≥ no and for all positive integer p

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Cauchy Sequences

The sequence (an) = (1/n)

For > 0, |an – am| = |1/n – 1/m| choose no to be any positive integer greater than 1/, we get |an – am| <

ILLUSTRATION

If = 0.01, no = 101 > 1/0.01 so that

|an – am| < 0.01 for all n, m ≥ no 22

Cauchy Sequences - Example

ILLUSTRATION Cont’d

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Cauchy Sequences - Example

n m |an – am|

101 102 0.000098

102 103 0.000093

105 106 0.000089

Revisit: The sequence (an) =((-1)n) is not a Cauchy sequence

Based on the equivalent form (p =1)

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Cauchy Sequences - Example

n allfor 2

odd isn if 11)(

even isn if 1)(1

11aa 1nn1nn

The sequence (an) = ((-1)n) cont’d

If < 2, it is not possible to find no such that |an – an+1| < for all n ≥ no. Hence, ((-1)n) is not CauchySimilarly (n) is not Cauchy

Since |n – m| > 1 if n m; for < 1 there does not exist any no 25

Cauchy Sequences - Example

• The behavior of Cauchy Sequence and convergence have to be defined based on the range

• Consider the sequence 1, 1.4, 1.41, 1.414, ... (convergent to the square root of 2) is Cauchy, but does not converge to a rational number

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Cauchy Sequences

A sequence may or may not converge

Even if it converges, “the limit” may not be easily visible

More detailed methods are needed to find the existence of a limit and its value – Algebra of limits

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Summary

A sequence may further be classified as Divergent, Oscillating SequenceCauchy sequences - we do not need to refer to the unknown limit of a sequenceIn effect both concepts are the same.

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Summary

(n/2n) is a convergent sequence

Evaluate the limit of i) ii)

(1/n2) is a Cauchy sequence

((-1)n n) is not a Cauchy sequence29

Questions

72n

43nn

1n