Infinite Sequences(Chapter 10) Module 1

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Learning OutcomeAt the end of the module, students able to

identify the formula or rule for a given infinite sequence.

identify whether a given infinite sequence converges or diverges.

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Infinite SequenceA infinite sequence or simply sequence ,is an unending succession of numbers, called terms in some definite order or pattern, 1st term3rd termnth term

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Example 1

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Infinite SequenceA formula or rule can be used to generatethe terms of an infinite sequence:Note: n represent natural numbers 1, 2, 3,

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Sequence notationIn general, terms of an infinite sequencecan be written as formula or rule withina bracket.

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Example 2

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Graph of SequenceBelow is graphical view of a sequence versuscontinuous curve:

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Graph of SequenceBelow is graphical view of a sequence:

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ExercisesExercise 10.1, Page 637- 6381,2,3,4

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Convergence/Divergenceof an infinite sequence

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Divergence of an infinite sequencen123456789101112345678f(n)The terms in the sequence, increase without bound.The sequence diverges

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Divergence of an infinite sequence

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Convergence of an infinite sequencen12345678910111234f(n)The terms in the sequencedecrease toward a limitingvalue of 0.The sequence converges to 0.

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Convergence of an infinite sequence (Illustration)(a)(b)

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Convergence/Divergenceof an infinite sequenceA sequence is said to converge to the limit, if otherwise the sequence is said to diverge.

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ExampleDetermine whether the sequence converges ordiverges. If it converges, find the limit.

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Example - solutionTherefore, the sequence converges and it

converges to

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Example 1 - solutionTherefore, the sequence

converges to since

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TheoremIf,thenTherefore, the sequence converges to

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Example 2 Determine whether the sequence converges ordiverges. If it converges, find the limit.

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Example 2 -solutionTherefore, the sequence

converges to

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Example 2 -solutionTherefore, the sequence

converges to

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Example 2 -solutionTherefore, the sequence diverges.

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Example 2 -solutionTherefore, the sequence converges to

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TheoremIf and are convergent sequences and c is a constant, then

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TheoremIf and are convergent sequences and c is a constant, then

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The Sandwich TheoremIf andthen

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Example Determine whether the sequenceconverges or diverges?

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Example - solution

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Example - solution By the Sandwich Theorem:

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Example - solution The sequenceconverges to 0.

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TheoremThe sequence is convergent if and divergent for all other values of r.

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Example Determine whether the sequenceconverges or diverges.

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Example - solution Since r = 2/3 < 1,

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Example - solution The sequenceconverges to 0.

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ExercisesExercise 10.1, Page 637-6385-30 ( odd numbers only)

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What is increasing sequence?A sequence {an} is called increasing if an < an+1 for allExample:

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What is decreasing sequence?A sequence {an} is called decreasing if an > an+1 for allExample

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What is monotonic sequence?A sequence that is either increasingor decreasing is called monotonic.

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What is bounded above sequence?A sequence {an} is said to be bounded aboveif there is a number M such that for all

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What is bounded below sequence?A sequence {an} is said to be bounded belowif there is a number m such that for all

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What is bounded sequence?A sequence {an} is said to be a bounded sequence if it is bounded above and below.

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Example 4Determine whether the given sequence is increasing,decreasing or non monotonic?

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Example 4(i) - solutionfor all n1, Why?If the nth term isthen the (n+1)th term is Therefore, the sequence is decreasing.

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Example 4(ii) - solutionfor all n 1Therefore, {bn} is a decreasing series.

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Monotonic Sequence TheoremEvery bounded, monotonic sequence isconvergent.IncreasingDecreasingor

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ExercisesExercise 10.2, Page 6451-23 (Odd number only)

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END OF THE MODULE 1

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A sequence to calculus is similar as calculator to scientist.In calculus Newtons discovered that functions can be represented or approximated as sums of infinite series.In some areas such optics, special relativity, electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represent it.These sequence may arise from data collected from experiments on regular basis.For example, the amount of rainfall per day.A sequence can also be defined as a function with positive integer domain.These sequence may arise from data collected from experiments on regular basis.For example, the amount of rainfall per dayNote: There are some sequences that do not have a simple defining formula or rule. For example: The amount of rainfall for each month Students ID numbers

The Fibonacci sequence {fn is defined recursively where each term is the sum of the two preceding termsi.e f1 = 1, f2 = 1, fn = fn-1 + fn-2 for n>=3The sequence {bn} is squeezed between the sequences {an} and {cn}.The sequence {bn} is squeezed between the sequences {an} and {cn}.The sequence {bn} is squeezed between the sequences {an} and {cn}.The sequence {bn} is squeezed between the sequences {an} and {cn}.The sequence {bn} is squeezed between the sequences {an} and {cn}.The sequence {bn} is squeezed between the sequences {an} and {cn}.