Lecture 16 :
Series (I) (10.2)
An infinite series (or simply a series) is a sum
of the form:
{z· · · + a}n + |an+1{z+ · ·}·
|a1 + a2 +
We express this sum in summation notation as
∞
X
an.
n=1
The nth partial sum of a series is given by
Sn = a1 + a2 + · · · + an
It is the sum of the first n terms of the series.
1
Sum of a Series : Let s = lim Sn,
n→∞
∞
X
n=1
an = lim Sn = s
n→∞
(The sum of the series is the limit of the nth partial sums.)
We say the series converges to s if s is a finite
number, otherwise, the series diverges.
2
ex. Given the sequence
3 3
3
3
,
{an} =
,
,
,··· =
10 100 1000
10n
what is the sum of the series
∞
∞ X
X
3
an =
=?
n
10
n=1
n=1
Solution:
S1 = 3/10 =
S2 = 3/10 + 3/100 =
S3 = 3/10 + 3/100 + 3/1000 =
..
SN = 3/10 + 3/100 + · · · + 3/10N =
lim SN =
N →∞
∞ X
3
= lim SN =
n
N →∞
10
n=1
3
ex. Suppose {an} = 1, determine the conver∞
X
gence of the series
an.
n=1
Difference between sequences and series:
• With a sequence, we consider the limit of
the individual terms an,
• With a series, we are interested in the sum
of the terms
a1 + a2 + a3 + · · ·
which is defined as the limit of the partial
sums.
Key Questions about series:
1. Does the series converge?
2. If the series converges, what is its sum?
4
A Geometric series is a series such that each
term is obtained by multiplying the previous term
by a fixed constant r, r 6= 0, that is, r is a common
ratio between 2 adjacent terms an+1/an = r.
∞
X
.
arn = a + ar + ar2 + · · ·
n=0
Sum of Geometric Series
∞
X
ar
A geometric series
arn =
, if |r| < 1
1−r
n=1
and diverges if |r| ≥ 1.
pf:
SN = ar + ar2 + · · · + arN
rSN =
ar2 + · · ·
+ arN +1
Important thing is: the numerator of this formula
is always the first term of the series.
5
∞
X
5
converges. If
ex. Determine if the series
n
2
n=3
converges, find the sum (Geo., )
5
4
If
P
an converges, then lim an = 0.
n→∞
nth-Term Test (for Divergence):
If
lim an 6= 0, then the series diverges.
n→∞
Warning: This test does NOT work in REVERSE. If lim an = 0, we can draw NO conn→∞
clusions about its convergence or divergence.
6
Harmonic Series
∞
X
1
n=1
n
Common Mistake Just because the
P ans decrease to 0 does NOT mean the series
an converges. The harmonic series diverges even though
its terms decrease to zero.
7
NYTI:Determine if the series converges or diverges. If converges, find the sum.
•
∞
X
n=1
n2
5n2 + 4
20
40
• 5 − 10
+
−
3
9
27 ....
•
∞
X
en
n=1
•
∞
X
n2
(−1)n
n=1
8
•
∞
X
(1/2)n−1
(Geometric, 2)
n=1
•
∞ X
1+
n=1
•
∞
X
1
n
n
(−5/4)n
(Geo., Diverges)
n=3
•
∞
X
22n31−n
n=1
∞
X
en
•
n−1
3
n=1
•
∞
X
√
n
2
(Geo.,
3e
3−e )
(TFD)
n=1
∞
X
2
n
•
(−1)n
2+1
11n
n=1
9
• A ball is droopped from a height of 6 feet and
begins bouncing. The height of each bounce is
3/4th the height of the previous bounce. Find
the total vertical distance traveled by the ball.
D = D1 + D2 + D3 + · · ·
= 6 + 2(6(3/4)) + 2(6(3/4)(3/4)) + · · ·
= 6 + 12(3/4) + 12(3/4)2 + 12(3/4)3 + · · ·
= 6 + 12(
∞
X
(3/4)n) = 42
n=1
10
NYTI (more):
1. If you delete a finite number of terms from a divergent series, will the new series still diverge?
explain your reasoning.
2. If you add a finite number of terms to a convergent series, will the new series still converge?
Explain your reasoning.
NOTE: Whether a series converges does not
depend on a finite number of terms added to
or removed from the series. However, the value
of a convergent series does change if nonzero
terms are added or deleted.
3. If lim an = 0, then
n→∞
∞
X
n=1
4. If |r| < 1, then
∞
X
arn =
n=1
5. The series
an converges. (T/F)
∞
X
n=1
a
. (T/F)
1−r
n
diverges. (T/F)
1000(n + 1)
11
6.
12