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MTH 253Calculus (Other Topics)
Chapter 11 – Infinite Sequences and Series
Section 11.2 – Infinite Series
Copyright © 2009 by Ron Wallace, all rights reserved.
Sums of the Terms of a Sequence
1
3
10nn
0.3, 0.03, 0.003, 0.0003, ...
Adding these terms gives …
10.3333...
3
1
9
10nn
What would be the sum of the terms of ?
Infinite Series
The sum of the terms of an infinite sequence is called an infinite series.
Notation:
1 2 3 41
...kk
a a a a a
NOTES:
• ak is some function of k whose domain is a set of integers.
• k can start anywhere (0 or 1 is the most common)
• The following are all the same: 1 1
k n nk n
a a a
Partial Sums of an Infinite Series
1 2 3 41
...kk
a a a a a
2 1 2s a a
3 1 2 3s a a a
1 2 31
...n
n n kk
s a a a a a
1 1s a
1n n
s
Sequence of partial sums.
●●●
Recursive Definition: 1n n ns s a
Converging/Diverging Series
1 2 3 41
...kk
a a a a a
1n n
s
If converges to ,S
then the series converges and1
kk
a S
If the sequence of partial sums diverges, then so does the series (it has no sum).
S is not often easy or even possible to determine!
Example …
1
1 1 1 1 1...
2 2 4 8 16kk
1
1
2s
2
1 1 3
2 4 4s
3
1 1 1 3 1 7
2 4 8 4 8 8s
4
1 1 1 1 7 1 15
2 4 8 16 8 16 16s
Pattern?2 1
2
n
n ns
2 1lim
2
n
nn
1
1lim 1
2nn
1
NOTE: A general expression for sn is usually difficult to determine.
Geometric Series
Each term is obtained by multiplying the proceeding term by a fixed constant.
1 2 3
1
...k
k
ar a ar ar ar
Example:
1
3 3 3 3 ...
10 10 100 1000kk
NOTE: w/ geometric series, k can start with any value (usually 0 or 1).
3
10a
1
10r
0
k
k
ar
Geometric Series
1
1 0
k k
k k
ar ar
• a is the value of the first term• r is the “common ratio”• r > 0, all terms have the same sign• r < 0, terms alternate signs
Geometric Series
Under what conditions does a geometric series converge?
1
1
k
k
ar
Case 1a: r = 1
1
1 1
...
k
k k
n
ar a a a a
s
an lim nn
s
Divergent!
Geometric Series
Under what conditions does a geometric series converge?
1
1
k
k
ar
Case 1b: r = -1
1 1
1 1
( 1) ...
k k
k k
n
ar a a a a
s
, if is odd
0, if is even
a n
n
lim nn
s
DNE
Divergent!
Geometric Series
Under what conditions does a geometric series converge?
1
1
k
k
ar
Case 2: |r| 1
2 3 1... nns a ar ar ar ar
2 3 4 ... nnrs ar ar ar ar ar
nn ns rs a ar
times
r
subtract
(1 )1 1
nn
n
a ar as r
r r
Geometric Series
Under what conditions does a geometric series converge?
1
1
k
k
ar
Case 2: |r| 1
Convergent if |r| < 1; Divergent Otherwise
(1 )1 1
nn
n
a ar as r
r r
lim nn
s
, if 1
1, if 1
ar
rr
Geometric Series
Determine the following sums, if they exists …
1
1
k
k
ar
, if 11
, if 1
ar
rr
3 32 412 6 3
1
3
4
k
k
0
4
3kk
27 27100 100004.272727... 4 +
Telescoping Series
1 1 2 2 3 31
( ) ( ) ( ) k kk
a b a b a b a b
1where n nb a
1n ns a b
1
Therefore, if ...
, then the series converges to lim
DNE or , then the series divergesnn
L a Lb
Telescoping Series - Examples
1
1 1
1k n n
Telescoping Series - Examples
1
6
(2 1)(2 1)k n n
Hint: “Partial Fractions”
1
3 3
(2 1) (2 1)k n n
ak is the kth term
sk is the kth partial sum
nth-Term Test
NOTE: p q implies that ~q ~p, but not ~p ~q or q p
If lim 0, then diverges.k kk
a a
If converges, then lim 0.k kk
a a
Proof …
1k k ka s s
1 1lim lim( ) lim lim 0k k k k kk k k k
a s s s s S S
The nth-Term Test (aka: The Divergence Test):
Algebraic Properties of Infinite Series
1k
k
a
1k
k
b
If … … are convergent …
… then …
1 1 1
( )k k k kk k k
a b a b
… are convergent.
&
NOTE: p q does NOT imply that q p or ~p ~q.
Algebraic Properties of Infinite Series
0c If
… then …
1 1 1
& k k kk k k
a ca c a
… are both convergent or both divergent.
Algebraic Properties of Infinite Series
0K If
… then …
1
& k kk k K
a a
… are both convergent or both divergent.
That is, a finite number of terms can be added to or removed from a series without affecting its convergence or divergence.
Algebraic Properties of Infinite Series
k k n k nk m k m n k m n
a a a
Example:
“Change of Index” or “Reindexing”
1 5
5
!k k
k
k
5( 4)
( 4)!
k
k
0k
5( 1)
( 1)!
k
k
