MATH115Sequences and Infinite Series

Paolo Lorenzo Bautista

De La Salle University

June 29, 2014

PLBautista (DLSU) MATH115 June 29, 2014 1 / 16

Sequences

DefinitionA sequence function is a function whose domain is the set

{1, 2, 3, . . . ,n, . . .}

of all positive integers.

Two sequences {an} and {bn} are said to be equal if ai = bi for allpositive integers i.

PLBautista (DLSU) MATH115 June 29, 2014 2 / 16

Sequences

DefinitionA sequence function is a function whose domain is the set

{1, 2, 3, . . . ,n, . . .}

of all positive integers.Two sequences {an} and {bn} are said to be equal if ai = bi for allpositive integers i.

PLBautista (DLSU) MATH115 June 29, 2014 2 / 16

Sequences

ExampleList down the first few elements of the following sequences:

1.{

nn + 1

}2.{

22n− 1

}

3. f (n) =

2 if n is odd2n + 1

if n is even

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Sequences

DefinitionA sequence {an} has the limit L if for any � > 0, there exists a numberN > 0 such that if n is an integer and

if n > N, then |an − L| < �.

This is written as limn→+∞

an = L, and we say that the sequence {an} isconvergent.

TheoremIf lim

x→+∞f (x) = L and f is defined for every positive integer, then

limn→+∞

f (n) = L, when n is restricted to positive integers.

PLBautista (DLSU) MATH115 June 29, 2014 4 / 16

Sequences

DefinitionA sequence {an} has the limit L if for any � > 0, there exists a numberN > 0 such that if n is an integer and

if n > N, then |an − L| < �.

This is written as limn→+∞

an = L, and we say that the sequence {an} isconvergent.

TheoremIf lim

x→+∞f (x) = L and f is defined for every positive integer, then

limn→+∞

f (n) = L, when n is restricted to positive integers.

PLBautista (DLSU) MATH115 June 29, 2014 4 / 16

Sequences

ExampleDetermine the limit of the following sequences, if they exist.

1.{

nn + 1

}2.{

22n− 1

}3.{

2n2 + 13n2 − n

}4.{

ln nn2

}5.{

1√n2 + 1− n

}6.{

n sin πn}

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Sequences

TheoremIf {an} and {bn} are convergent sequences and c is a constant, then

i) the constant sequence {c} has c as its limit;ii) lim

n→+∞can = c lim

n→+∞an;

iii) limn→+∞

an ± bn = limn→+∞

an ± limn→+∞

bn;

iv) limn→+∞

anbn =(

limn→+∞

an

)(lim

n→+∞bn

);

v) limn→+∞

anbn

=lim

n→+∞an

limn→+∞

bnif lim

n→+∞bn 6= 0 and every bn 6= 0.

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Sequences

ExampleDetermine the limit of the following sequences, if they exist.

1.{

4n3

2n2 + 1sin πn

}2.{

(2n2 + 1)n3n2 − n

sin πn

}

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Sequences

DefinitionA sequence {an} is

i) increasing if an ≤ an+1 for all n;ii) decreasing if an ≥ an+1 for all n.

A sequence is monotonic if it is either increasing or decreasing.

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Sequences

ExampleDetermine if the following sequences are increasing, decreasing, or notmonotonic.

1.{

nn + 1

}2.{

22n− 1

}3.{

(−1)n+1

n

}

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Sequences

ExerciseDetermine if the following sequences are increasing, decreasing, or notmonotonic.

1.{

n2 − 1n

}2.{

2n

1 + 2n

}3.{

(2n)!5n

}

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Sequences

DefinitionThe number C is a lower bound of the sequence {an} if C ≤ an for allpositive integers n, and the number D is an upper bound of thesequence {an} if an ≤ D for all positive integers n.

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Sequences

DefinitionIf A is a lower bound of a sequence {an} and if A has the property thatfor every lower bound C of {an}, C ≤ A, then A is the greatest lowerbound of the sequence. Similarly, if B is an upper bound of a sequence{an} and if B has the property that for every upper bound D of {an},D ≥ B, then B is the least upper bound of the sequence.

PLBautista (DLSU) MATH115 June 29, 2014 12 / 16

Sequences

DefinitionA sequence is bounded if and only if it has an upper bound and alower bound.

Theorem (Axiom of Completeness)Every nonempty set of real numbers that has a lower bound has agreatest lower bound. Also, every nonempty set of real numbers thathas an upper bound has a least upper bound.

TheoremA bounded monotonic sequence is convergent.

PLBautista (DLSU) MATH115 June 29, 2014 13 / 16

Sequences

DefinitionA sequence is bounded if and only if it has an upper bound and alower bound.

Theorem (Axiom of Completeness)Every nonempty set of real numbers that has a lower bound has agreatest lower bound. Also, every nonempty set of real numbers thathas an upper bound has a least upper bound.

TheoremA bounded monotonic sequence is convergent.

PLBautista (DLSU) MATH115 June 29, 2014 13 / 16

Sequences

DefinitionA sequence is bounded if and only if it has an upper bound and alower bound.

Theorem (Axiom of Completeness)Every nonempty set of real numbers that has a lower bound has agreatest lower bound. Also, every nonempty set of real numbers thathas an upper bound has a least upper bound.

TheoremA bounded monotonic sequence is convergent.

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Sequences

ExampleUse the previous theorem to prove that the following sequences areconvergent.

1.{

3n− 14n− 5

}2.{

1 · 3 · 5 · · · · · 2n− 12 · 4 · 6 · · · · · 2n

}3.{

5n

1 + 52n

}4.{

n!1 · 3 · 5 · · · · · 2n− 1

}5.{

n2

2n

}

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Sequences

TheoremLet {an} be an increasing sequence, and suppose that D is an upperbound of this sequence. Then {an} is convergent, and

limn→+∞

an ≤ D.

TheoremLet {an} be a decreasing sequence, and suppose that C is an lowerbound of this sequence. Then {an} is convergent, and

limn→+∞

an ≥ C.

PLBautista (DLSU) MATH115 June 29, 2014 15 / 16

Sequences

TheoremLet {an} be an increasing sequence, and suppose that D is an upperbound of this sequence. Then {an} is convergent, and

limn→+∞

an ≤ D.

TheoremLet {an} be a decreasing sequence, and suppose that C is an lowerbound of this sequence. Then {an} is convergent, and

limn→+∞

an ≥ C.

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Sequences

TheoremA convergent monotonic sequence is bounded.

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Sequences