Chapter 10. Infinite sequences and series
Section 10.2 Series
An expression of the form
a1 + a2 + ... + an + ... =
∞
X
an
n=1
is called an infinite series or series.
Consider partial sums:
s 1 = a1 ,
s 2 = a1 + a2 ,
.................................
sn = a1 + a2 + ... + an
Definition Given a series
∞
P
an , and let Sn =
n=0
n
P
ak . If the sequence {sn }∞
n=1 converges
k=1
and lim sn = s, then the series is called convergent and we write
n→∞
∞
X
an = s
n=0
The number s is called the sum of the series. Otherwise, the series is called divergent.
The geometric series
( a
∞
X
, if |r| < 1
arn =
1−r
∞,
if |r| ≥ 1
n=0
Example 1. Determine whether the series is convergent or divergent. If it is convergent, find
its sum.
∞
X
4n+1
(a)
5n
n=1
1
(b)
∞
X
n=1
1
n(n + 2)
Example 2. Write the number 0.307 as a ratio of integers.
∞
X
1
The harmonic series
is divergent.
n
n=1
∞
X
1
The p-series
is convergent for p > 1 and divergent for p ≤ 1
np
n=1
∞
P
Theorem. If
an is convergent, then lim an = 0.
n→∞
n=1
If lim an = 0, we can not conclude that
n→∞
∞
P
an is convergent.
n=1
Test for divergence If lim an does not exist or lim an 6= 0, then
n→∞
n→∞
2
∞
P
n=1
an is divergent.
Example 3. Show that
∞
P
arctan n is divergent.
n=1
Theorem, If
c is a constant),
∞
P
an and
n=1
∞
P
(an + bn ),
n=1
(i)
∞
P
can = c
n=1
(iii)
∞
P
∞
P
bn are convergent series, then so are the series
n=1
∞
P
∞
P
n=1
(an − bn ), and:
n=1
an
n=1
(an − bn ) =
n=1
∞
P
∞
P
(ii)
(an + bn ) =
n=1
∞
P
n=1
an −
∞
P
∞
P
n=1
an +
∞
P
bn
n=1
bn
n=1
NOTE. A finite number of terms can not affect the convergence of the series.
∞
X
3n + 2n
.
Example 4. Find the sum of the series
n
6
n=1
3
can (where