Section 9.2 – Series and Convergence

Section 9.2 – Series and Convergence

Goals of Chapter 9

• Approximate Pi• Prove infinite series are another important

application of limits, derivatives, approximation, slope, and concavity of functions.

• Find challenging antiderivatives like • Lay the groundwork for future courses

Summation NotationA compact notation (often called sigma notation) for sums

is the following:

1 21

n

i ni

a a a a

Upper Limit of Summation

General Term

Lower Limit of Summation

Index of Summation

Examples

Evaluate: 3

2

1

4 1i

i

24 1 1 i=1

24 2 1 i=2

24 3 1 i=3

3 15 35

53

Series investigate the following:

2

1

4 1i

i

Infinite Sum

1

1 What is the area of the square?

square unitCut the square in half and label

the area of one section.𝟏𝟐

Cut the unlabeled area in half and label the area of one

section.𝟏𝟒 Continue the process…

𝟏𝟖 𝟏

𝟏𝟔

𝟏𝟑𝟐 𝟏

𝟔𝟒

𝟏𝟏𝟐𝟖

Sum all of the areas:

1 1 1 1 1 1 12 4 8 16 32 64 128 ... 1

2...n

The general term is…1

Since the infinite sum represents the area of the square…

12

1n

n

Infinite Series

An infinite series is an expression of the form

, or

The numbers are the terms of the series; is the nth term.

Connecting Series and SequencesConsider the Series:

Find the sum of…The first term: The first 2 terms:The first 3 terms:The first 4 terms:

Consider the sequence of PARTIAL SUMS above:

The sequence of PARTIAL SUMS appear to converge to:

Partial Sums of a SeriesThe partial sums of the series for a sequence:

of real numbers, each defined as a finite sum.

Convergent or Divergent Series

If the sequence of partial sums has a limit as , we say the series converges to the sum , and we write

Otherwise, we say the series diverges.

ExamplesInvestigate the partial sums of the sequences below to determine if the series converges or diverges. If it converges, state the limit.

1. 2. 3. 4. 5. 6.

¿𝟏−𝟏+𝟏−𝟏+𝟏−𝟏+…

¿𝟎 .𝟓+𝟎 .𝟎𝟓+𝟎 .𝟎𝟎𝟓+𝟎 .𝟎𝟎𝟎𝟓+…¿𝟏+𝟑+𝟓+𝟕+…¿𝟎 .𝟏+𝟎 .𝟐+𝟎 .𝟒+𝟎 .𝟖+…

¿𝟓𝟎+𝟐𝟓+𝟏𝟐 .𝟓+𝟔 .𝟐𝟓+…

𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔¿𝟓𝟗

𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔

¿𝟏𝟎𝟎

Why do 1, 3, 4 and 6 Diverge? The limit of the general term does not equal 0.

¿𝟑+𝟐 .𝟓+𝟐 .𝟑𝟑+𝟐 .𝟐𝟓+…𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔

The n-th Term TestIf , then the infinite series diverges.

OR

If the infinite series converges, then .

Is the converse of this statement true?If , does the infinite series always converge?

When determining if a

series converges, always use this

test first!

The Converse of The n-th Term Test

Consider the two famous sequences below:

1

1 1 1 1: 1 ... ...2 3 4

1 1 1 1: 1 ... ( 1) ...2 3 4

n

HarmonicSeries n

AlternatingHarmonic Series n

For both series’, the . BUT do both series’ converge?Check a calculator program.

1lim 0n n

1 1lim( 1) 0n

n n

The Converse of The n-th Term Test

11 11 1

: : ( 1)nn nn n

Harmonic AlternatingSeries Harmonic Series

The Alternating Harmonic Series appears to converge to ~0.69.

The Harmonic Series appears to diverge.

The n-th Term TestIf , then the infinite series diverges.

OR

If the infinite series converges, then .

The converse of this statement is NOT true.If , the infinite series does not necessarily converge.

When determining if a

series converges, always use this

test first!

The Harmonic Series DivergesProve the Harmonic Series diverges:

Compare the Series to the graph of .

……

Find the Left Hand Riemann Sum to approximate .

1 12 1

3 14

15 ...

The Left Hand Riemann Sum is equal to the Sum of the

Harmonic Series.



…

So…

Since is decreasing, the Left Hand Riemann Sum is an over estimate.

Thus:



…

So… We can find the value of the

improper integral:

Since diverges and , the Harmonic Series Diverges.

11

limb

xbdx

1lim ln

b

bx

lim ln ln1b

b

The Harmonic Series Diverges Part 2

Justify that the Harmonic Series diverges another way:

Investigate the sum:121 1 1

3 4 1 1 1 15 6 7 8 1 1

9 16... ...12

12 1

2 12

By increasing the size of , we can make the sum of the infinite series as large was we desire.

The Alternating Harmonic Series Converges

Justify the Alternating Harmonic Series converges:

1 12

13

14

15

0 12

1

Investigate and plot the sum:16

17 ...

The sum is bounded by 0.5 and 1.

Each Successive term in the sequence of partial

sums is between the two previous terms in

this sequence .

The sum must be between any two successive terms.

?S

We will find the actual value of the

sum soon.

Arithmetic and Geometric SeriesAn Arithmetic Series has a constant difference between terms. (Similar to an Arithmetic Sequence.)Example:

A Geometric Series has a constant ratio between terms. (Similar to a Geometric Sequence.)Example:

Arithmetic and Geometric SeriesBy the n-th Term Test, every Arithmetic Series diverges:

Some Geometric Series diverge and others converge:

Since Geometric Series occasionally converge, we will focus on them.

Definition of a Geometric SeriesIn a geometric series each term is obtained from its preceding term by multiplying by the same number :

Examples: The previous examples are geometric.

∑𝑛=1

∞ 510𝑛

∑𝑛=1

∞

0.1(2)𝑛 ∑𝑛=1

∞

100 (0.5)𝑛

White Board ChallengeFind the general term and the sum of the first 10 terms of the sequence:

18 1.5 nna

10 906.641s

Finite Sum of a Geometric SeriesFind the sum of the first terms of a geometric series:

2 3 1... nS a ar ar ar ar 2 3 1... n nrS ar ar ar ar ar ________________________________

nS rS a ar 1 1 nS r a r

1

1

na rS

r

What happens to the sum as the value of n increases to infinity?

Multiply by r.

Subtract the two equations.

Solve for the sum.

Check with the

previous example.

Infinite Sum of a Geometric Series

Consider :

1if r 1if r 1

1limna r

rnS

Diverges

1

1limna r

rnS

1 01lim arn

1ar

Depends on the value of r.

Convergent Geometric Series

The geometric series converges if and only if . If the series converges, its sum is .

Example: Find the sum if it exists.1. 2. 3.

Where a is the first term and r is the constant ratio.

0.3a 0.3 11 .1 3S

0.030.3 0.1r

1125 2.5a 2.5

1 0.5 5S 12r

2a 12 /4/2 2r

Diverges

Documents

Section 9.2 – Series and Convergence