Chapter 9.2Chapter 9.2Chapter 9.2Chapter 9.2

SERIES AND CONVERGENCESERIES AND CONVERGENCE

After you finish your HOMEWORK you will be able to…

• Understand the definition of a convergent infinite series

• Use properties of infinite geometric series

• Use the nth-Term Test for Divergence of an infinite series

INFINITESERIES

• An infinite series (aka series) is the sum of the terms of an infinite sequence.

• Each of the numbers, , are called terms of the series.

1 2 31

... ...n nn

a a a a a

na

CONVERGENT AND DIVERGENT SERIES

For the infinite series , the n-th partial sum is given by .

If the sequence of partial sums, , converges

to , then the series converges. The limit is called the sum of the series.

If diverges, then the series diverges.Series may also start with n = 0.

1n

n

a

1 2 31

...n

n n nn

S a a a a a

S 1n

n

a

S

nS1 2 3 ... ...nS a a a a

nS

THE BATHTUB ANALOGY

DIVERGE VERSUS CONVERGE

Consider the series

What happens if you continue adding 1 cup of water?

Consider the series

How is this situation different?

Will the tub fill?

1

1n

1

1

2nn

TELESCOPING SERIESWhat do you notice about the

following series?

What is the nth partial sum?

1 2 2 3 3 4 4 5a a a a a a a a

CONVERGENCE OF A TELESCOPING SERIES

A telescoping series will converge if and only if approaches a finite number as n approaches infinity.

If it does converge, its sum is

na

1 1lim nn

S a a

GEOMETRIC SERIESThe following series is a geometric series

with ratio r.

2

0

, 0n n

n

ar a ar ar ar a

THEOREM 9.6CONVERGENCE OF A GEOMETRIC SERIES

A geometric series with ratio r diverges if .

If then the series converges to

1r

0 1,r

0

, 0 1.1

n

n

aar r

r

THEOREM 9.7PROPERTIES OF INFINITE SERIES

If is a real number,

then the following series converge to theindicated sums.

1 1

, , n nn n

a A b B c

1

n nn

a b A B

1n

n

ca cA

1

n nn

a b A B

THEOREM 9.8LIMIT OF nth TERM OF A CONVERGENT SERIES

If converges, then

Why?

1n

n

a

lim 0.n

na

The nth-Term TestTHEOREM 9.9

If , the

infinite series

diverges.

lim 0nna

1n

n

a

1

Example:

1

n

nn

nS

n