Chapter 9.2 Chapter 9.2 SERIES AND CONVERGENCE SERIES AND CONVERGENCE
Jan 03, 2016
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