Math 1B
§ 11.2 Series
∞
An infinite series is an expression that can be written in the form
The numbers a1,a2 ,a3 ,... are called the terms of the series.
∑a
n
= a1 + a2 + a3 + ...+ an + ...
n=1
Since it is impossible to add infinitely many numbers together, sums of infinite series are defined and
€ consider the infinite decimal
computed by an indirect limiting process. To motivate this,
€
0.3333…
Which can be viewed as an infinite series
0.3 + 0.03 + 0.003 + 0.0003 + …
=
3
3
3
3
+ 2 + 3 + 4 + ...
10 10 10 10
1
Since 0.3333… is the decimal expansion of , any reasonable definition for the sum of an infinite
3
€
1
series should give as the sum.
3
Let’s build a sequence {s1,s2 ,s3 ,...}€in the following way:
€
€
€
3
s1 = = 0.3
10
3€ 3
s2 = + 2 = 0.33
10 10
3
3
3
s3 = + 2 + 3 = 0.333
10 10 10
3
3
3
3
s4 = + 2 + 3 + 4 = 0.3333
10 10 10 10
€
….
€
sn =
3
3
3
3
+ 2 + 3 + ... + n
10 10 10
10
….
What does the sequence {s1,s2 ,s3 ,...} converge to? Let’s calculate the limit of the general term sn .
3
3
3
3
lim sn = lim + 2 + 3 + ... + n
n → ∞ 10
10 10
10
€
n →∞
€
Use the following trick to find this limit:
1
sn =
10
sn −
1
sn =
10
9
sn =
10
sn =
So
lim sn =
n →∞
Note: This is not really a sum but shorthand for a limit statement.
We are now ready to define the general concept of the “sum” of an infinite series.
We consider the partial sums
s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
...
n
sn = a1 + a2 + a3 + ...+ an = ∑ ai
i=1
∞
The sequence {sn }n=1 is called the sequence of partial sums.
€
∞
Definition:
€
Given a series
∑a
n
= a1 + a2 + a3 + ..., let sn denote its nth partial sum:
n=1
n
sn = ∑ ai = a1 + a2 + a3 + ...+ an
i=1
If the sequence {sn }€is convergent and lim sn = S exists, then the series
n →∞
write
€
€
∑a
n
is convergent and we
∞
∑a
n
n=1
=S
or
a1 + a2 + a3 + ... + an + ... = S
€
The number S is called the sum of the series. Otherwise, the series is called divergent.
€
n
∞
The sum of a series is the limit of the sequence of partial sums:
∑ an = lim ∑ ai = lim sn = S
n=1
∞
Example: Consider the infinite series
n→∞
i=1
n→∞
1
∑ n(n +1)
n=1
a) Find the first four terms of the sequence of partial sums.
b) Find an expression for sn and make a conjecture about the sum of the series.
∞
Example: Is the series
∑ (−1)
n= 0
€
n
convergent or divergent? If it is convergent, find its sum.
To evaluate an infinite series, it is necessary to determine a formula for the sequence of partial sums
{sn } and then find its limit. This procedure can be carried out with two types of series we will look at
in this section: geometric series and telescoping series.
€
Geometric Series
An important example of an infinite series is the geometric series
∞
2
3
a + ar + ar + ar + ...+ ar
n−1
+ ... = ∑ ar n−1
a≠0
n=1
where each term is obtained by multiplying the preceding one by the common ratio r.
€ Theorem:
∞
∑ ar
The geometric series
n −1
= a + ar + ar 2 + ...
n =1
is convergent if r < 1 and its sum is
∞
∑ ar
n −1
=
n =1
a
1− r
r <1
If r ≥ 1 , the geometric series is divergent.
Some examples of geometric series:
Example: Find the sum of the geometric series
5−3+
9 27 81
−
+
− ...
5 25 125
∞
Example: Is the series
∑9
−n +2
4 n +1 convergent or divergent? If it is convergent, find its sum.
n=1
€
Example: Write the number 0.784 as a ratio of integers.
Geometric series are a rare exception to a general rule: often it is not very hard to show that a series
converges, but finding what it converges to is extremely difficult.
There is one more exception to this rule: telescoping sums.
Telescoping Series
Example: Is the series
∞
∑n
n =1
2
1
convergent or divergent? If it is convergent, find its sum.
+ 3n + 2
With geometric series and telescoping series, the sequence of partial sums can be found and its limit
can be evaluated (when it exists). Unfortunately, it is difficult or impossible to find an explicit formula
for the sequence of partial sums for most infinite series. Therefore, it is difficult to obtain the exact
value of most convergent series.
In light of these observations, we now shift our focus and ask a simple yes or no question: Given an
infinite series, does it converge?
∞
Property:
The harmonic series
1
∑n
is divergent.
n=1
Theorem:
∞
If the series ∑ an is convergent, then lim an = 0
n→∞
€n =1
Note: If lim an = 0 , we can’t conclude that
n→∞
The Test for Divergence:
∑a
n
is convergent.
If lim an does not exist of if lim an ≠ 0 , then the series
n→∞
n→∞
∞
∑a
n
is
n =1
divergent.
Example: Is the series
4n 2 − n 3
convergent or divergent?
∑
3
n =1 10 + 2n
∞
∑ a is divergent. But if lim a
convergence or divergence of ∑ a .
Note: If lim an ≠ 0 , we know
n→∞
n
n
€
€
n→∞
n
= 0 , we know nothing about the
Theorem:
∑ (a
n
i)
€
∑a
n
and
∑b
n
are convergent series, then so are the series
− bn ) , and
∞
∞
n=1
∞
n=1
∑ ca€n = c ∑ an €
ii)
€
If
∞
∑ (a
n
n=1
∞
iii)
n
n=1
∞
+ bn ) = ∑ an + ∑ bn
n=1
∞
∑ (a
€
n=1
∞
− bn ) = ∑ an −∑ bn
n=1
n=1
€
∞
€
Example: Find the sum of the series
n=1
€
#
∑%$ n
2
4
3&
− n(
+ 3n + 2 4 '
€
∑ ca , ∑ (a
n
n
+ bn ) , and