ad calculus 3
DESCRIPTION
Sequences and Series of real numbersTRANSCRIPT
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Lecture - 3
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Cauchy Sequences
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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!
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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•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
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•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
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•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!
11
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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
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•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
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•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
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(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
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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