Download Testing Convergence at Endpoints

Section 9.5b
Alternating Series
A series in which the terms are alternatingly positive and
negative is an alternating series.
Examples:
1 1 1
2  1    
2 4 8
1


2
n
n
4

This is a geometric series with a = –2 and r = –1/2
So it converges to:
4
2
2


3
1   1 2  3 2
Alternating Series
A series in which the terms are alternatingly positive and
negative is an alternating series.
Examples:
1 2  3  4  5  6 
  1
n 1
n
This series diverges by the nth-Term Test:
lim  1
n 
1 1 1 1
1    
2 3 4 5
n 1
n

1


n
n 1

This is the alternating harmonic series, and it converges by
a new test that we will see shortly…
The Alternating Series Test (Leibniz’s Thm)

The series
  1
n 1
n 1
un  u1  u2  u3  u4 
converges if all three of the following conditions are satisfied:
1. each un is positive;
 un 1 for all n  N, for some integer N ;
3. lim un  0 .
2. un
n 
The Alternating Series Estimation Thm

n 1
If the alternating series  n 1  1 un satisfies the
conditions of Leibniz’s Theorem, then the truncation error
for the nth partial sum is less than un 1 and has the same
sign as the first unused term.
The Alternating Series Test (Leibniz’s Thm)
Closing in on the sum of a convergent alternating series:
u1
u2
u3
u4
0
S2
S4
L
S3
S1
x
This figure not only proves the fact of convergence; it also
shows the way that an alternating series converges. The
partial sums keep “overshooting” the limit as they go back
and forth on the number line, gradually closing in as the
terms tend to zero.
The Alternating Series Test (Leibniz’s Thm)
Prove that the alternating harmonic series is convergent,
but not absolutely convergent. Find a bound for the
truncation error after 99 terms.
The terms are strictly alternating in sign and decrease in
absolute value from the start:
1 1
1  
2 3
.
Also,

By the Alternating Series Test,

n 1
1
lim  0
n  n
 1
n
n 1
converges.
The Alternating Series Test (Leibniz’s Thm)
Prove that the alternating harmonic series is convergent,
but not absolutely convergent. Find a bound for the
truncation error after 99 terms.

However, the series
1

n 1 n
of absolute values is the harmonic series, which we know
diverges, so the alternating harmonic series is not
absolutely convergent.
The Alternating Series Estimation Theorem guarantees
that the truncation error after 99 terms is less than
1
1
u991 

99  1 100
Absolute and Conditional Convergence
Because the alternating harmonic series is convergent but
not absolutely convergent, we say it is conditionally
convergent (or converges conditionally).
We take it for granted that we can rearrange the term of
a finite sum without affecting the sum. We can also
rearrange a finite number of terms of an infinite series
without affecting the sum. But if we rearrange an
infinite number of terms of an infinite series, we can be
sure of leaving the sum unaltered only if it converges
absolutely.
Practice Problems
Determine whether the given series converges absolutely,
converges conditionally, or diverges. Give reasons for your
answer. 
n 1 1  n
  1
n 1
1 n
un  2
n
n
2
is a decreasing series with positive terms, and
lim un  0
n 
So the series converges by the Alternating Series Test.
1 n

2
n 1 n

Check for absolute convergence:
Practice Problems
Determine whether the given series converges absolutely,
converges conditionally, or diverges. Give reasons for your
answer. 
n 1 1  n
  1
n 1
1 n

2
n 1 n
n

behaves like

1

n 1 n
2
1
n
for large values of n…
diverges, so
1 n

2
n
n 1

also diverges…
Conclusion:
The original series converges conditionally
Practice Problems
Determine whether the given series converges absolutely,
converges conditionally, or diverges. Give reasons for your
answer. 
n 1 sin n
  1
n 1

Take a look at:

n 1
n
2
sin n
2
n
So this series also converges
by direct comparison
is always less than
or equal to:

1

2
n 1 n
Which we know converges
as a p-series with p = 2.
Conclusion:
The original series converges absolutely
Intervals of Convergence

How to Test a Power Series
 c  x  a  for Convergence
n 0
n
n
1. Use the Ratio Test to find the values of x for which the
series converges absolutely. Ordinarily, this is an open
interval
a  R  x  a  R.
In some instances, the series converges for all values of x.
In rare cases, the series converges only at x = a.
2. If the interval of convergence is finite, test for
convergence or divergence at each endpoint. The Ratio
Test fails at these points. Use a comparison test, the
Integral Test, or the Alternating Series Test.
Intervals of Convergence

How to Test a Power Series
 c  x  a  for Convergence
n 0
n
n
3. If the interval of convergence is a  R  x  a  R ,
conclude that the series diverges (it does not even
converge conditionally) for x  a  R , because for those
values of x the nth term does not approach zero.
Finally, take a look at the flowchart on p.505,
and read the “Word of Caution”…
Practice Problems
Find (a) the interval of convergence of the series. For what values of
x does the series converge (b) absolutely, (c) conditionally?

  x  5
n
n 0
This is a geometric series which converges only for
x 5 1
 6, 4 
(b)  6, 4 
(a)
(c) None
1  x  5  1
6  x  4
Practice Problems
Find (a) the interval of convergence of the series. For what values of
x does the series converge (b) absolutely, (c) conditionally?


n 1
 3x  2 
n
n
Ratio Test for absolute convergence:
n 1
3x  2
an 1
 lim
lim
n 
n  a
n 1
n
n
3x  2
The series converges absolutely when
n
 3x  2
3x  2  1
1  3 x  2  1
1
Now, we check the endpoints…
 x 1
3
Practice Problems
Find (a) the interval of convergence of the series. For what values of
x does the series converge (b) absolutely, (c) conditionally?


 3x  2 
n
Check
n
n 1

Check
(a)
(b)

n 1
 1
n
n
converges conditionally
n
1
x  1: 
n 1 n
1 
,1
 3 
1
x :
3

diverges
1 
 ,1
3 
(c) At
1
x
3
Practice Problems
Find (a) the interval of convergence of the series. For what values of
x does the series converge (b) absolutely, (c) conditionally?


n 0
x
2 n 1
n!
Ratio Test for absolute convergence:
2 n 3
2
x
an 1
n!
x
 lim
lim

lim

0
2
n

1
n   n  1 !
n  a
n  n  1
x
n
This series converges absolutely for all real numbers.
(a) All real numbers
(b) All real numbers
(c) None