Section 10.2: Series
Def: A series a sum of a sequence, that is
∞
P
an =a1 + a2 + a3 + ... + a100 + ... + .... The
n=1
goal is to determine when the series has a finite sum.
Def: If
∞
P
an = S, where S is finite, then we say the series converges and its sum is S. If
n=1
∞
P
an = ∞ or does not exist, then we say the series diverges. Moreover,
n=1
∞
X
an = lim
n→∞
n=1
1. Discuss the convergence/divergence of
∞
P
n.
n=1
2. Discuss the convergence/divergence of
∞
P
1.
n=1
3. Discuss the convergence/divergence of
∞
P
(−1)n .
n=1
4. Discuss the convergence/divergence of
∞ 1
P
.
n
n=1 2
n
X
i=1
ai
Def: Consider the series
∞
P
an = a1 + a2 + a3 + ... + an+1 + ....
n=1
We will construct the sequence of partial sums as follows:
{sn } = {s1 , s2 , s3 , ..., ...}, where
s1 = a1 Called the first partial sum
s2 = a1 + a2 Called the second partial sum
s3 = a1 + a2 + a3 Called the third partial sum
Therefore a general formula for sn , the nth term of the sequence, is
sn =
n
P
ai = a1 + a2 + ... + an Called the nth partial sum
i=1
Note: For series that do not begin at an index of one, for example
∞
P
an , we call s1 = a2
n=2
the first partial sum, s2 = a2 + a3 the second partial sum, etc. Thus in general, the nth
partial sum is the sum of the first n terms, regardless of where the series begins.
Moreover,
∞
P
n=1
an = lim sn = lim
n→∞
n
P
n→∞ i=1
ai
5. Find the first 6 terms in the sequence of partial sums for
s1 =
s2 =
s3 =
s4 =
s5 =
s6 =
∞ 1
P
.
n
n=1 2
1
X
1
1
= = 0.5
i
2
2
i=1
2
X
1
1
1
= + 2 = 0.75
i
2
2
2
i=1
3
X
1
1
1
1
= + 2 + 3 = 0.875
i
2
2
2
2
i=1
4
X
1
1
1
1
1
= + 2 + 3 + 4 = 0.9375
i
2
2 2
2
2
i=1
5
X
1
1
1
1
1
1
= + 2 + 3 + 4 + 5 = 0.96875
i
2
2
2
2
2
2
i=1
6
X
1
1
1
1
1
1
1
= + 2 + 3 + 4 + 5 + 6 = 0.984375
i
2
2 2
2
2
2
2
i=1
∞
P
6. Find the first 5 terms in the sequence of partial sums the series
(1). From this, what can
n=1
be concluded about the convergence/divergence of this series?
7. Find the first 5 terms in the sequence of partial sums the series
∞
P
(−1)n . From this, what
n=1
can be concluded about the convergence/divergence of this series?
8. Find the first 5 terms in the sequence of partial sums the series
∞
P
sin(nπ). From this, what
n=1
can be concluded about the convergence/divergence of this series?
Test for Divergence: If lim an 6= 0, then
n→∞
∞
P
an diverges. Intuitively, this makes sense
n=1
because if lim an 6= 0, then we are continuing to add terms together that are not getting
n→∞
∞
P
smaller, hence
an must diverge. NOTE: The converse is not necessarily true: If
n=1
lim an = 0, then the series
n→∞
∞
P
an does not necessarily converge. Therefore if you
n=1
find that lim an = 0, then the test for divergence fails and thus another test must be applied.
n→∞
9. Use the Test For Divergence to determine whether the series diverges:
∞
P
arctan(n)
a.)
n=1
b.)
∞
P
(1 + (−1)n )
n=2
10. Explain why the test for divergence fails when applied to the series
∞
P
n=1
n2
n
+1
Finding the sum of a series. Recall if {sn } is the sequence of partial sume of the series
∞
∞
P
P
an , then
an = lim sn . In other words, the sum of a series is the limit of the sequence
n=1
n=1
n→∞
of partial sums.
11. If the nth partial sum of the series
∞
P
n=1
a.) The sum of the first 5 terms.
an is sn =
n+1
, find:
2n + 4
b.) The sum of the first 20 terms.
c.) Does the series converge or diverge? If it converges, what is the sum?
d.) What is a1 ? What is a10 ?
e.) Find a general formula for an .
12. If the nth partial sum of the series
∞
P
an is sn = e1/n , does the series converge? Support
n=1
your answer.
Def: A TELESCOPING SERIES is a series of the form
∞
P
(an+i − an ) for some integer
n=1
i ≥ 1. Telescoping series can be identified by expanding the sum to see if an infinate number
of terms cancel.
13. Determine whether the following telescoping series converges or diverges. If it converges, find
the sum.
∞
P
1
1
sin − sin
a.)
n
n+1
n=1
b.)
∞
P
n=7
4
4
−
n+2 n+4
c.)
∞
P
n=1
d.)
ln
n+1
n+2
7
n=1 n(n + 2)
∞
P
Def: A GEOMETRIC SERIES is a series of the form
∞
X
arn−1 = a + ar + ar2 + ar3 + ... + arn−1 + arn + ...
n=1
The geometric series
∞
P
arn−1 will converge if |r| < 1 and will diverge if |r| ≥ 1. Moreover,
n=1
if |r| < 1, then
∞
P
arn−1 =
n=1
a
.
1−r
Note that not all series begin at 1, and not all powers of r are n − 1. No matter
the situation, a is always the firt term of the series.
∞
P
Proof: Let’s form the sequence of partial sums for the series
arn−1 .
n=1
sn =
n
P
ari−1 = a + ar + ar2 + ar3 + ... + arn−1
i=1
rsn =
n
P
ari = ar + ar2 + ar3 + ar4 + ... + arn
i=1
sn − rsn = a − arn
sn (1 − r) = a(1 − rn ), thus sn =
Now,
∞
P
a(1 − rn )
.
1−r
a(1 − rn )
a
=
, provided |r| < 1.
n→∞
1−r
1−r
an = lim sn = lim
n=1
n→∞
14. Determine whether the following geometric series converge or diverge. If it converges, find
the sum. If it diverges, explain why.
n
∞
P
2
a.)
5
7
n=1
b.)
∞ −4
P
n
n=0 e
c.)
23n
n+1
n=0 (−5)
d.)
∞ (−1)n + 3n
P
4n
n=1
∞
P
e.) 9 + 2 +
f.)
8
4
+
+ ...
7 49
∞ 5
P
n
n=3 2
15. Consider
∞
P
(x − 5)n . Find the value(s) of x for which the series converges. Find the sum of
n=1
the series for those values of x.
Summary
Note: So far, there are 3 types of series whose sum we can find:
a.) One where the formula for the nth partial sum, sn , is given, in which case
∞
P
an = lim sn .
n=1
n→∞
b.) Geometric
∞
P
arn−1 =
n=1
a
for |r| < 1.
1−r
c.) Telescoping (When the series is expanded, terms cancel). Find a formula for sn , then use
∞
P
the fact that
an = lim sn .
n→∞
n=1
Let’s see what you have learned.
1. True or false: If
∞
P
an converges, then lim an = 0.
n=1
2. True or false: If sn =
lim an = 0.
n→∞
∞
P
n
is the sum of the first n terms of the series
an , then
n+1
n=1
n→∞
3. True or false:
∞ n
P
is a geometric series.
n
n=1 3
n+1
2
2
, r = . Does the series converge?
4. True or false: For the geometric series
(−1)
3
3
n=2
If, what is the sum of the series?
∞
P
n
2n
.
3n + 4
a.) Determine whether {an } is convergent
5. Let an =
b.) Determine whether
∞
P
an is convergent.
n=1
6. Discuss the convergence/divergence of the following series:
a.)
1
5
+
2−n
n=1
∞
P
b.) The series
∞
P
an , where the sequence of partial sums is:
n=1
(i) sn =
c.)
∞
P
n=1
n2 + 1
n
cos(nπ)
(ii) sn = 1 +
1
n
7. Find the sum of the series
∞
P
n=1
2
3
.
+
n2 + n 3 n