Lecture - 3

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Cauchy Sequences

Contents

• Subsequences• Cauchy’s Convergence Criterion

• CCC• Infinite Series

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SubSequence• Let (an) be a sequence. Let (nk) be a

strictly increasing sequence of natural numbers. Then (ank

) is called a subsequence of (an)• (a2n) is a subsequence of any sequence

(an)• Any sequence is a subsequence of itself

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SubSequence• A subsequence of a convergent

sequence, converges to the same limit

• Every sequence has a monotonic subsequence

• Every bounded sequence has a convergent subsequence

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SubSequence• If (an) = 1,0,1,0…is oscillating finitely

(a2n) = 0, 0, ….. is a convergent subsequence of (an)

• If (an) = 1,-2,3,-4…is oscillating infinitely

• (a2n) = -2, -4, ….. → -∞

• (a2n-1) = 1, 3, ….. → ∞

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• A sequence (an) is said to converge to l if given > 0, there exists a m N such that |an - l| < for all n ≥ m.

• 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

p N 6

Revive – A Little!

Any Convergent Sequence is a Cauchy sequence

For > 0,

|an – am| = |an – a + a – am|

≤ |an – a| + |a – am| which could be made less than

(since (an) a )7

Road map to CCC

Any Cauchy sequence is bounded. From the result, | |an| – |am| | < |an – am|

Then for a Cauchy sequence, there exists no N,

| |an| – |am| | < |an – am|

< for all n > no

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Road map to CCC

Cont’d. Hence - < |an| – |am| < ,

|an| < |am| +

If k = max{|a1|, |a2|, |a3|…. |am| + } then |an| ≤ k

Hence (an) is bounded

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Road map to CCC

If a Cauchy sequence (an) has a subsequence (ank

)→a, then (an)→a

For > 0, |an - am| < /2 for all n,m ≥ no Since (an) is Cauchy

Also, |ank- a| < /2 for all k ≥ ko

Since (ank

)→a

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Road map to CCC

Choose nk such that nk ≥ nko and no

|an - a| = | an - ank + ank

- a|

< | an – ank | + |ank

- a|

< for all n ≥ no

Hence (an) → a11

Road map to CCC

Cauchy Sequence Convergence A Cauchy sequence (an) is bounded

It has a converging subsequence (ank

) → a

Hence, (an) → a

CCC: (an) in R is convergent

(an) is Cauchy

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Road map to CCC

Consider a sequence

(an) = 1+1/2+….+1/n

If (an) → a by CCC it is Cauchy

|an – am| < ½ n, m ≥ no

(by considering = ½ )

Now consider |a2m – am|

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CCC - Illustration

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CCC - Illustration

2

1

2m

1m.

2m

1...

2m

1

2m

1

2m

1...

2m

1

1m

1

m

1...

2

11

2m

1...

2

11

2

mmaa

Let (an) = a1, a2,……an… be a sequence of real numbers.

The expression a1+a2+…+an +……. is called an infinite series of real numbers

Symbol:

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Series of Positive terms

n

1nn

aor a

Let s1 = a1

s2 = a1+a2

s3 = a1+a2+a3 and so on

sn= a1+a2+……+an

Then (sn) is called the sequence of partial sums of the given series.

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Series of Positive terms

•The series is said to Converge,

Diverge or Oscillate according as the sequence (sn) Converges, Diverges or Oscillates

•If (sn) → s, we say that converges to the sum s•The behavior of a series does not change if a finite number of terms are added or altered

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Infinite Series a

1nn

a1n

n

•The series 1 + 1 + 1 +……………

s1 = 1

s2 = 2

s3 = 3 sn= n

(sn) = (n), this sequence diverges to ∞

Hence, the given series diverges to ∞

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Infinite Series - Example

•The series

s1 = 1

s2 = sn=

(sn) → e, and hence the given series is converging to e

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Infinite Series - Example

n!

1........

3!

1

2!

1

1!

11

1!

11

1)!-(n

1........

3!

1

2!

1

1!

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•The series 1+1/2+….+1/n+……

s1 = 1

s2 = 1+1/2

s3 = 1+ 1/2+1/3

sn= 1+1/2+….+1/n

(sn) →∞

Hence, the given series diverges to ∞20

Infinite Series - Example

•The Geometric series

1+ r + r2 +…+ rn ……

sn= 1+ r + r2 +…+ rn =

•(rn) → 0 when 0 ≤ r < 1

•(rn) → ∞ when r >1

•When r =1 (sn)= (n) → ∞21

Infinite Series - Example

r1

r1 n

•The Geometric series – Cont’dsn= 1+ r + r2 +…+ rn =

•r = -1 then sn is 0 or 1 (n: even / odd)

(sn) oscillates finitely

•When r < -1 (rn) oscillates infinitely 22

Infinite Series - Example

r1

r1 n

(an) in R is convergent (an) is CauchyUseful in convergence Infinite SeriesSequence of partial sums define the convergence of an infinite series

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Summary

Try CCC to study the convergence of(1/n), ((-1)n), (n)

Test the convergence of 1 + 2 + …….

Test the convergence of

Test the convergence of -1 + 2 - 3+…. (Use CCC)

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Questions

2

11n

n