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9.2Series and Convergence
If we add all the terms of a sequence, we get a series:
1 2 31
n kk
a a a a a
a1, a2,… are terms of the series. an is the nth term.
To find the sum of a series, we need to consider the partial sums:
1 1S a
2 1 2S a a
3 1 2 3S a a a
1
n
n kk
S a
nth partial sum
If Sn has a limit as , then the series converges,
otherwise it diverges.
n
Series
Examples
1n
n
0
)1(n
n
2
22 1n
n
n
Determine whether the series is convergent or divergent.
Divergence Test
If then the series diverges.lim 0nn a 1
nn
a
Examples: Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1 25
13
n n
n
1
1
)3
(n
n
nn
e
1
1
1n n n
Using partial fractions:
1 A 0 A B
0 1 B
1 B
1
1 1
1n n n
1 1 1 11 1
2 3 3 42
1 1...
1
nS
n n
11
1nSn
lim 1nnS
1
11
A B
n n nn
1 1A n Bn
1 An A Bn
Example
in which nearly every term cancels with a preceding or following term. However, it doesn’t have a set form. Partial fraction decomposition is often used to put in the
above form. Partial sum will be considered since most terms can be
canceled.
Example:
Telescoping Series
...)()()( 433221 bbbbbb
A telescoping series is any series that can be written in the following (or similar) form
12 158
1
n nn
This infinite series converges to 1.
1
11
2
1
21
4
1
4
1
8
1
8
1
161
16
1
32 1
64
1
32
1
64 1
Example
1 2
1
nn
Determine whether the series is convergent or divergent.
In a geometric series, each term is found by multiplying the preceding term by the same number, r.
2 3 1 1
1
n n
n
a ar ar ar ar ar
This converges to if , and diverges if .1
a
r1r 1r
1 1r is the interval of convergence.
Geometric Series
Examples
Determine whether the series is convergent or divergent.
...12
1
4
1
8
1
11
1
5
)6(
nn
n
.3 .03 .003 .0003 .333... 1
3
310
11
10
a
r
3109
10
3
9
1
3
Example: Write 3.545454… as a rational number.
1 4
3
5
2
nn
n
n