Lecture -1

Dr. M. Subbiah, M. Sc M. Phil Ph. D

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Contents

• Real Sequences

• Bounded Sequences

• Monotone of Sequences

• Principle of Convergence

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Sequences

• Let f: N → R be a function and let f(n) = an Then a1, a2, …..an….is called the sequence in R determined by the function and is denoted by (an)

• an is called the nth term of the sequence

• The range of the function f, which is a subset of R is called the range of the sequence

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Sequences - Example

1.f: N → R given by f(n) = n determines the sequence 1, 2, 3,…..

2.f: N → R given by f(n) = (-1)n determines the sequence -1, 1, -1, 1, -1,……

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Sequences - Example

3. f: N → R given by f(n) = 5 determines a constant sequence 5,5,5,5,……

4. f: N → R given by f(n) =

determines the sequence

1/2, 2/3, 3/4,…… 5

1+nn

Sequences - Example

5. f: N → R given by f(n) = xn-1 (x∈R) determines the geometric sequence 1,x,x2….

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Bounded Sequences

• A sequence (an) is said to be bounded above if there exists a real number k s.t an ≤ k for all n ∈ N. Then k is called an upper bound (UB) of (an).

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Bounded Sequences

• A sequence (an) is said to be bounded below if there exists a real number k s.t an ≥ k for all n ∈ N. Then k is called a lower bound (LB) of (an).

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Bounded Sequences

•A sequence (an) is said to be bounded if it is both bounded above and below. That is |an| ≤ k (k ≥ 0)

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Bounded Sequences – Examples

1.The sequence 1,1/2,1/3….1/n……...

a LB: 0 (any k < 0) an UB:1 (any k > 1)

2. The sequence 1, 4, 9, 16….n2…….

a LB: 1 (k < 1) not bounded above

3. The sequence -1, -2, -3, ….-n…….

not bounded below an UB: -1 (k > -1)10

Bounded Sequences – LUB, GLB

4. The sequence 0,1,-1,2 -2,3 -3, …n, -n……not bounded below, not bounded above • Bounds may or may not exist.

Even if exist, they need not be a member of the sequence

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Bounded Sequences – LUB, GLB

• Let u ∈ R. u is called the Least Upper Bound (LUB) or Supremum (sup) of a sequence (an)

(i) if u is an upper bound of (an) (ii) If v < u, then v is not an upper bound of (an)

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Bounded Sequences – LUB, GLB

• Let l ∈ R. u is called the Greatest Lower Bound (GLB) or infimum (inf) of a sequence (an) if

(i) l is an upper bound of (an) (ii) If m > l, then m is not a

lower bound of (an)

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Bounded Sequences – Examples1. The sequence 1,1/2,1/3….…….

a LB: 0 (any k < 0) an UB:1

(any k > 1)LUB: 1 and GLB: 0

1. The sequence 1, 4, 9, 16….…….

a LB: 1 (k < 1) not bounded

above GLB : 1 no LUB

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Bounded Sequences – Examples

3. The sequence -1, -2, -3, ….….

not bounded below an UB: -1 (k > -1)LUB : -1 no GLB

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Monotonic Sequences

• A sequence (an) is said to be monotonic increasing (↑) if an ≤ an+1 for all n ∈ N.

• A sequence (an) is said to be monotonic decreasing (↓) if an ≥ an+1 for all n ∈ N.

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Monotonic Sequences

• A sequence (an) is said to be monotonic if it is either monotonic increasing or monotonic decreasing

• If equality is removed from above two definitions, then (an) is said to be strictly ↑ or strictly ↓ sequence accordingly 17

Monotonic Sequences• 1,2,2,3,3,3,4,4,4,4…..is monotonic

increasing (↑) where as 1,2,3,….is strictly monotonic increasing

• 1, 1/2, 1/3, is strictly decreasing ↓ sequence

• The sequence ((-1)n) is not monotonic

• is a monotonic increasing

sequence (may not be obvious) 18

23n

72n

+−

Monotonic Sequences

19n allfor aa Hence, 0 aa So

5)2)(3n(3n

25

233n

7 22n

23n

72n

21)3(n

7 1)2(n

23n

72n aathen

23n

72n a If

1nn1nn

1nn

n

++

+

<<−++

−=

++−+−

+−=

++−+−

+−=−

+−=

Monotonic Sequences

• A monotonic increasing sequence (an) is bounded below and a1 ≤ a2 ≤ a3 ≤ … a1

GLB of (an)

• A monotonic decreasing sequence (an) is bounded above and a1 ≥ a2 ≥ a3 ≥… a1

LUB of (an)

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Monotonic SequencesHowever, UB of ↑ or LB of ↓ is inconclusive

•00

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Sequence LB UB

1,2,3…..is ↑ but not bounded above Obvious: 1 No

-1,-1/2,-1/3 is ↑ but bounded above Obvious: -1 0

-1,-2,-3…..is ↓ but not bounded below No Obvious: -1

1,1/2,1/3 is ↓ but bounded below 0 Obvious: 1

Summary•Sequence is a real valued function on N

•Lower / Upper bound may exist for a sequence

•A monotonic Sequence is if an ≤ an+1 or an ≥ an+1 for all n ∈ N

•Always ↑ sequence is bounded below and monotonic ↓ sequence is bounded above

•UB of ↑ or LB ↓ of is inconclusive

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Questions• Write the first 10 terms of the sequence

• Are the sequences (logn) and monotonic?

• Find LUB and GLB if it exists (1+n+n2) and 1/2,2/3,3/4…..

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−

n10

11

3

2 and

+ 22 xn

cosnx

−+

12

122

2

n

n