138 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
an excellent resource and, given different learning styles, somewhat personal: you may want to
maintain your own annotated list.
4.2.1 TAILS OF SEQUENCES
You may have noticed that we are a little careless with where we start the index of summation
on our infinite series. is is because, while the value of a convergent series depends on every
term, the convergence or divergence behavior does not.
Definition 4.11 If we take a sequence and make a new sequence by discarding a finite number of
initial terms, the new sequence is a tail of the old sequence.
Knowledge Box 4.17
Tail convergence
A sequence converges if and only if all its tails converge.
e practical effect of Knowledge Box 4.17 is that, if a few initial terms of a sequence are
causing trouble, you may discard them and test the remainder of the sequence to determine
convergence or divergence.
Example 4.34 Determine the convergence of the series:
1
X
nD0
.1/
n
n
2
6n C 10
Solution:
If we look at the first several terms of this series we get:
1
10
1
5
C
1
2
1 C
1
2
1
5
C
1
10
1
17
C
e series alternates signs, getting larger in absolute value for the first four terms, but then
getting smaller in absolute value for the remaining terms. is means that the alternating series
test works for the tail of the sequence starting at the fourth term. Remember that the alternating
series test requires that the terms shrink in absolute value.
4.2. SERIES CONVERGENCE TESTS 139
We conclude that the series converges by the alternating series test applied to a tail of
the sequence.
˙
e next example shows a very special sort of series for which we can compute the exact value.
Once you understand how these series work, you can construct many examples of them. You
may want to review partial fractions from Fast Start Integral Calculus.
Example 4.35 Prove that the sequence
1
X
nD0
1
n
2
C 3n C 2
converges to exactly 1.
Solution:
We can see, by limit comparison to
1
X
nD1
1
n
2
, a convergent p-series, that this series converges.
Knowing the exact value of the sum is another matter. at will require a little algebra.
1
X
nD0
1
n
2
C 3n C 2
D
1
X
nD0
1
.n C1/.n C 2/
D
1
X
nD0
A
n C1
C
B
n C2
-partial fractions
D
1
X
nD0
1
n C1
1
n C2
D 1 1= 2 C 1=2 1=3 C 1=3 1=4 C 1=4 1=5 C
D 1 C 0 C 0 C 0 C 0 C
D 1
˙
140 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
e series in Example 4.35 is called a telescoping series, in analogy to the way a small telescope
or spyglass collapses for storage. In essence, the series is composed of a positive and a negative
series that cancel out all but one term.
Knowledge Box 4.18
Telescoping series
If b
n
D a
n
a
nC1
then
1
X
nD0
b
n
D a
0
:
When expanded in terms of the a
k
values, everything except a
0
can-
cels out.
Trace out the information in Knowledge Box 4.18 for Example 4.35.
e next Knowledge Box gives a guide for choosing a convergence test. As with choosing the
best method for integration, the only real way to get better at choosing the correct convergence
test is to work examples. Lots of examples.
Knowledge Box 4.19
Choosing a convergence test
1. If the general term of the series fails to converge to zero, then it
diverges by the divergence test. Remember that this does not work
in reverse – if the general term converges to zero, anything might
happen.
2. If the series is a geometric series or a p-series, use the tests for those
series.
3. If the general term of the series is a
n
D f .n/ for a function f .x/
that you can integrate, try the integral test.
4. A positive series that is, term-by-term, no larger than a series that
converges, converges (comparison test) – look for this.
continued
4.2. SERIES CONVERGENCE TESTS 141
Knowledge Box 4.20
Choosing a convergence test—Continued
5. Similarly, a positive series that is, term-by-term, no smaller than
a divergent series diverges, again by the comparison test.
6. If you have a series that looks like a series you know how to deal
with, try the limit comparison test. is is useful for things like
slightly modified p series or letting you use simpler integral tests.
7. You can often set up a comparison test or limit comparison test by
doing algebra or arithmetic to the general term of a series.
8. Remember that if the term-by-term absolute value of a series con-
verges, then the series converges. is is the absolute convergence
test.
9. If the terms of a series alternate in sign, look at the alternating series
test.
10. If you can take the limit of adjacent terms of a series in a reasonable
way, then the ratio test is a possibility.
11. If you can take the nth root of the general term of a series in a rea-
sonable way, the root test is a possibility.
12. If some finite number of initial terms of a series are preventing you
from using a test, tail convergence says you can ignore them and
then do your test.
13. Unless you’re taking a test or quiz, asking for advise and suggestions
is not the worst possible option.
142 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
PROBLEMS
Problem 4.36 For each of the following series, determine if the series converges or diverges.
State the name of the test you are using.
1.
1
X
nD1
1
n
e
2.
1
X
nD0
n C1
n
2
C 1
3.
1
X
nD0
n C1
5n C4
4.
1
X
nD1
ln
1
n
2
5.
1
X
nD0
2
n
C 1
3
n
C 1
6.
1
X
nD0
p
n
n
2
C 1
Problem 4.37 For each of the following series, determine if the series converges or diverges.
State the name of the test you are using.
1.
1
X
nD0
n
2
2
n
C 1
2.
1
X
nD0
0:0462
n
3.
1
X
nD2
sin.n/
n
p
2
4.
1
X
nD1
1
3
p
n
5.
1
X
nD1
.1/
n
3
p
n
6.
1
X
nD0
1
.2n/Š
Problem 4.38 For each of the following series, determine if the series converges or diverges.
State the name of the test you are using.
1.
1
X
nD1
1
n
n=2
2.
1
X
nD1
5
n
n
n=2
3.
1
X
nD1
e
n
4.
1
X
nD1
e
n
n
5.
1
X
nD1
e
2n
n
6.
1
X
nD1
e
3
C
n
n
2
Problem 4.39 Give an example to demonstrate that the divergence test does not work in re-
verse, i.e., a sequence whose general term goes to zero but whose sum is infinite.
Problem 4.40 Use the integral test to prove that the p-test works.
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