§12.6 –Ratio Test, Root Test, Absolute Convergence
Mark Woodard
Furman U
Fall 2010
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
1 / 10
Outline
1
Absolute convergence
2
The ratio test
3
The root test
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
2 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
Mark Woodard (Furman U)
P
an converges.
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
P
an converges.
Proof.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
P
an converges.
Proof.
We have −|an | ≤ an ≤ |an |; thus, 0 ≤ an + |an | ≤ 2|an |.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
P
an converges.
Proof.
We have −|an | ≤ an ≤ |an |; thus, 0 ≤ an + |an | ≤ 2|an |.
P
Thus the series (an + |an |) converges by SCT.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
P
an converges.
Proof.
We have −|an | ≤ an ≤ |an |; thus, 0 ≤ an + |an | ≤ 2|an |.
P
Thus the series (an + |an |) converges by SCT.
P
By hypothesis, the series
−|an | converges.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Theorem
If
P
|an | converges, then
P
an converges.
Proof.
We have −|an | ≤ an ≤ |an |; thus, 0 ≤ an + |an | ≤ 2|an |.
P
Thus the series (an + |an |) converges by SCT.
P
By hypothesis, the series
−|an | converges.
Consequently,
X
an =
X
(an + |an |) − |an |
converges as well.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
3 / 10
Absolute convergence
Problem
Examine the convergence of the following series:
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
4 / 10
Absolute convergence
Problem
Examine the convergence of the following series:
∞
X
sin(n)
n=1
n2
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
4 / 10
Absolute convergence
Problem
Examine the convergence of the following series:
∞
X
sin(n)
n=1
∞
X
n2
n+2
(−1)n+1 √
n5 + 8
n=1
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
4 / 10
Absolute convergence
Definition
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
5 / 10
Absolute convergence
Definition
P
P
If the series
|an | converges, then we say that the series
an
converges absolutely.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
5 / 10
Absolute convergence
Definition
P
P
If the series
|an | converges, then we say that the series
an
converges absolutely.
P
P
if the series
a
converges
but
|an | diverges, then we say that the
n
P
series
an converges conditionally.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
5 / 10
Absolute convergence
Problem
Does the series
∞
X
(−1)n+1
n=1
n
converge absolutely, converge conditionally,
or diverge?
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
6 / 10
Absolute convergence
Problem
Does the series
∞
X
(−1)n+1
n=1
n
converge absolutely, converge conditionally,
or diverge?
Solution
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
6 / 10
Absolute convergence
Problem
Does the series
∞
X
(−1)n+1
n=1
n
converge absolutely, converge conditionally,
or diverge?
Solution
The series of absolute values diverges by PST, p = 1.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
6 / 10
Absolute convergence
Problem
Does the series
∞
X
(−1)n+1
n=1
n
converge absolutely, converge conditionally,
or diverge?
Solution
The series of absolute values diverges by PST, p = 1.
The series converges by the AST.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
6 / 10
Absolute convergence
Problem
Does the series
∞
X
(−1)n+1
n=1
n
converge absolutely, converge conditionally,
or diverge?
Solution
The series of absolute values diverges by PST, p = 1.
The series converges by the AST.
Thus the series converges conditionally.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
6 / 10
The ratio test
Theorem (The Ratio Test (RT))
Let {an } be a sequence of nonzero real numbers and suppose that
|an+1 |
→L
|an |
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
7 / 10
The ratio test
Theorem (The Ratio Test (RT))
Let {an } be a sequence of nonzero real numbers and suppose that
|an+1 |
→L
|an |
If L < 1, then
Mark Woodard (Furman U)
P
an converges absolutely.
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
7 / 10
The ratio test
Theorem (The Ratio Test (RT))
Let {an } be a sequence of nonzero real numbers and suppose that
|an+1 |
→L
|an |
If L < 1, then
P
an converges absolutely.
If L > 1, then
P
an diverges.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
7 / 10
The ratio test
Theorem (The Ratio Test (RT))
Let {an } be a sequence of nonzero real numbers and suppose that
|an+1 |
→L
|an |
If L < 1, then
P
an converges absolutely.
If L > 1, then
P
an diverges.
If L = 1, then the test is inconclusive; the series may or may not
converge.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
7 / 10
The ratio test
Problem
Determine the convergence or divergence of the following series:
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
8 / 10
The ratio test
Problem
Determine the convergence or divergence of the following series:
∞
X
2n
n=1
n!
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
8 / 10
The ratio test
Problem
Determine the convergence or divergence of the following series:
∞
X
2n
n=1
∞
X
n=1
n!
n!
nn
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
8 / 10
The ratio test
Problem
Determine the convergence or divergence of the following series:
∞
X
2n
n=1
∞
X
n=1
n!
n!
nn
∞
X
1
n2
n=1
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
8 / 10
The ratio test
Problem
Determine the convergence or divergence of the following series:
∞
X
2n
n=1
∞
X
n=1
n!
n!
nn
∞
X
1
n2
n=1
∞
X
1
n=1
n
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
8 / 10
The root test
Theorem (The Root Test)
Suppose |an |1/n → L.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
9 / 10
The root test
Theorem (The Root Test)
Suppose |an |1/n → L.
P
If L < 1, then
an converges absolutely.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
9 / 10
The root test
Theorem (The Root Test)
Suppose |an |1/n → L.
P
If L < 1, then
an converges absolutely.
P
If L > 1, then
an diverges.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
9 / 10
The root test
Theorem (The Root Test)
Suppose |an |1/n → L.
P
If L < 1, then
an converges absolutely.
P
If L > 1, then
an diverges.
If L = 1, then the test is inconclusive; the series may or may not
converge.
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
9 / 10
The root test
Problem
n
∞ X
3n
converges.
Determine whether the series
5n + 6
n=1
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
10 / 10
The root test
Problem
n
∞ X
3n
converges.
Determine whether the series
5n + 6
n=1
Solution
Since
3n
3
→ < 1,
5n + 6
5
the series converges by the Root Test.
|an |1/n =
Mark Woodard (Furman U)
§12.6 –Ratio Test, Root Test, Absolute Convergence
Fall 2010
10 / 10