BC Calculus

Sec 11.2

Series

Sequence vs. Series

Sequence:

A list of terms

Series:

The SUM of a sequence

1 2, ,... ...na a a

1 2 ... na a a

Our primary concern for the next 2 weeks:

Does a given series (sum) converge or diverge?

If it converges, what does it converge to?

Topics for today

Terminology & Notation & Properties

Test for Divergence (aka “nth term test”)

Geometric Series

Harmonic Series

Telescoping Test

Sequence vs Series

Notation

1Given n na s a

n

The sum of the first 4

elements of sequence a.4

1 1 11

2 3 4s

4

1The 4th element of sequence

4a a

Summation Notation

1

n

n k

k

s a

k is a counter. It starts at the lower number, 1 in this

case, and goes to the upper number, n in this case.

They are similar to the bounds of an integral.

na

1k 2k 3k k n

3a2a1a

4

4

1

1 1 1 1 1

1 2 3 4k

sk

The letters k and n are arbitrary. They can be anything.

The starting and stopping values of our counter can be

anything.

1

n

j

j

a

8

3

m

m

a

1

n

n

a

0

n

n

a

Examples

Partial Sums, aka finite series

11 as

212 aas

3213 aaas

nn aaaas

generalIn

...

:

321

…

Infinite Series

Partial sums add a finite number of elements.

An infinite series adds an infinite number of elements.

1 2 1

1

n n n

n

a a a a a

Convergence

We are most interested in if a series (sum) converges

to a specific number or not

A partial sum (aka finite series) always converges.

An infinite sum may converge or diverge.

(similar to improper integrals)

Properties of Series

1 1

n n

n n

ca c a

1 1 1

n n n n

n n n

a b a b

1 1 1

b

n n n

n n n b

a a a

2 2

1 1

1 13 3

1 1n nn n

1 1 1

1 1 1 1

1 3 1 3n n nn n n n

10

1 1 11

sin sin sin

n n n

n n n

n n n

Constants

Sums and Differences

Piecewise

These properties can be used to make it easier to

determine the behavior of the series.

General Process

• Analyze problem and choose most appropriate test

• Do mechanics of the test

• Interpret results

• If necessary, start again with a different test

All series we study can be analyzed.

So, for every type of test we discuss:

• Must know when a test can and cannot be used

• Know how to actually do the test

• Know how to interpret results, what the test can and

cannot tell you and when it is inconclusive

Test for Divergence

Aka

nth term test

“Test for Divergence”

aka “nth term test”

If limn

an does not exist or lim

na

n 0

then the series an

n1

diverges.

When can you use it:

Always. Normally the first test you do.

If the individual terms of the SEQUENCE are not

approaching zero, the SERIES diverges. Period.

If the terms do approach zero, the test is

inconclusive. More tests are required.

Use and interpret the

nth term test

2

21

2

1

21

21

2

3

2

2

2

4 3

n

n

n

n

n n

n

n n

n

n

n n

n n

2

2

2lim 1 Sequence converges to nonzero #

3

Series diverges

n

n n

n

2 2

lim Sequence diverges

Series also diverges

n

n n

n

2lim 0 Sequences converges to 0

2

Series may converge or diverge

n

n

n n

2

2lim 0 Sequences converges to 0

4 3

Series may converge or diverge

n n n

Geometric Series

One type of series you

are familiar with is:

The Geometric series

3 6 12 24 48

1 2 3 43 3 2 3 2 3 2 3 2

1 3 2a r

1 1

1

1 1

3 2n n

n n

a r

Most common forms of

geometric series

a ar ar 2 ar3 ... ar n1 ... an1

r n1

or an0

r n

The key clue

If the base is a constant and n is in the

exponent, it is most likely geometric.

1

1

3

1

1 1

2 2

1 1 1

4 4 4

1 1

2 2

n

n

n

nn

n

n

r

r

r

Let’s look at

convergence

What if r =1?

...a a a a

In this case, the series diverges.

Now let’s consider a general geometric

series where r does not = 1

132 ... n

n ararararas

nn

n arararararrs 132 ...

Mult. by r

Now, subtract the 2 eq. n

nn ararss

n

n arars )1(

1

(1 ) 1

nn

n

a ra ars

r r

1

(1 )

n

n

a rs

r

lim n

nr

(1 )lim lim

(1 )

n

nn n

a rs

r

lim(1 )

nn

as

r

If –1 < r < 1, then as n increases w/out bound 0

A Geometric Series

Is convergent if 1r

and its sum is 1

1

a

r

If 1r , the geometric series is divergent

Ex:

1

1

26

3

n

n

1

26,

3a r

6converges to 18

1 (2 / 3)

Beware

1

1

140

2

n

n

1

140

2

n

n

1

1

140 20

2

1

2

a

r

1 40

1

2

a

r

What are a and r?

Non-obvious

Geometric Series

1

21

2

5

n

nn

The bases are constants,

The exponents have n’s,

It is geometric.

1

131

2

5 5

n

nn

1

1

1 2

125 5

n

n

1 2

125 5 a r

What you should do

1

21

2

5

n

nn

You do NOT need to rewrite

the series. That takes time and

is prone to error.

What is being raised to some power of n ?2

5

What is the initial value? 1 1

1 2 3

2 1 1

5 5 125

a

You Try

2

13

6

7

n

nn

r

a

6

7

3 2

3 1 4

6 6 6

7 7 2401

Another Geometric

Incognito

2

1

2

3

n

n

2

1

2

3

n

n

1

4

9

n

n

4 4

9 9r a

You try.

What are a and r?

3

1

3

5

n

n

1

27

125

n

n

27 27

125 125r a

A little more

complicated

2 1

1

1

2

n

n

2

1

1 1

2 2

n

n

1 1

4 8r a

1

1 1

2 4

n

n

You try.

What are a and r?

3 2

1

1

3

n

n

32

1

1 1

3 3

n

n

1 1

27 243r a

1

1 1

9 27

n

n

Another disguise

1

cos3

n

n

1

1

2

n

n

1 1

2 2r a

What is the common

thread?

1

21

2

5

n

nn

2

1

2

3

n

n

2 1

1

1

2

n

n

1

cos3

n

n

They all have n in the exponent and

Constants in the base. That is the key.

Harmonic Series

Definition

1

1 1 1 1Harmonic Series = 1 ...

2 3 4n n

The name “Harmonic series” comes from the

world of music and overtones, or harmonics.

The wavelengths of the overtones of a

vibrating string are

Source: Wikipedia.com

1 1 1, , ...

2 3 4

Does the harmonic series

converge or diverge?

1

1 1 1 1Harmonic Series = 1 ...

2 3 4n n

1limn n

Try the nth term test.

0

It is not a geometric series.

Try the geometric test.

We have to get creative.

1

1 1 1 1 1...

1 2 3 4n n

We’re going to do a comparison test.

If a sum is clearly less than a series that is

known to converge then

If a sum is clearly greater than a series that

is known to diverge then

The sum must also converge.

The sum must also diverge.

First, let’s group the terms.

1

1 1 1 1 1...

1 2 3 4n n

...16

1...

10

1

9

1

8

1

7

1

6

1

5

1

4

1

3

1

2

11

Now, this ...16

1...

10

1

9

1

8

1

7

1

6

1

5

1

4

1

3

1

2

11

Clearly exceeds:

1 1 1 1 1 1 11

2 4 4 8 8 8 8

1 1 1... ...

16 16 16

...2

1

2

1

2

1

2

11 Which equals:

Which clearly increases forever and diverges.

So, must diverge.

1

1

n n

So, the Harmonic

Series Diverges.

How fast?

1000

1

1

n n

1 million

1

1

n n

7.4851 1 1

12 3 1000

1 billion

1

1

n n

14.357

21

How many terms are

needed?

???

1

1100

n n

4310 terms required

Pick any positive # and, eventually, the

Harmonic series will surpass it.

Telescoping Test

(Last test of the day)

21

2

4 3n n n

Try the term test.thn

2

2lim 0

4 3n n n

Test is inconclusive.

It is not geometric.

It is not the harmonic series.

Telescoping Test

When to use it:

When the nth term test is inconclusive AND

You can rewrite the rule with partial fractions.

21 1

2

4 3 1 3n n

A B

n n n n

Doing the Telescoping

Test

21 1

2 1 1

4 3 1 3n nn n n n

1 1

1 1

1 3n nn n

Do the partial

fraction work.

Rewrite as two

series

21 1

2

4 3 1 3n n

A B

n n n n

Now, expand each

series

1

1

1

1

1

3

n

n

n

n

Conclusion

21

2 5 converges to

4 3 6n n n

Summary

Terminology

Series rules

Test for Divergence (nth term test)

Geometric Series

Harmonic Series

Telescoping Test