4.2. SERIES CONVERGENCE TESTS 143
Problem 4.41 Suppose that p.x/ is a polynomial. Use the integral test to demonstrate that
1
X
nD0
p.n/e
n
converges.
Problem 4.42 Suppose that q.x/ is a polynomial with exactly three roots, all of which are
negative real numbers. Demonstrate that
1
X
nD0
1
q.n/
converges.
Problem 4.43 If x
n
and y
n
are the general terms of a convergent series, then x
n
C y
n
are as
well. is requires only simple algebra. What is startling is that the reverse is not true. Find an
example of a
n
D x
n
C y
n
so that
1
X
nD1
a
n
converges, but neither of
1
X
nD1
x
n
or
1
X
nD1
y
n
converge.
Problem 4.44 Show that, when you apply the ratio test to a geometric series, the limit that
appears in the test is the ratio of the series.
Problem 4.45 Show that, when you apply the root test to a geometric series, the limit that
appears in the test is the ratio of the series.
Problem 4.46 Suppose r > 1. Prove that
1
X
nD0
n
k
r
n
converges when k is an integer 1.
144 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
Problem 4.47 Suppose that we are testing the convergence of a series
1
X
nD0
p.n/
q.n/
where p.x/ and q.x/ are polynomials. If we add the assumption that q.x/ has no roots at any
of the values of n involved in the sum, explain in terms of the degrees of the polynomials when
the series converges.
Problem 4.48 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
3
.1 Cn/.2 C n/.3 Cn/.4 C n/
2.
1
X
nD0
n
2
.1 Cn/.2 C n/.3 Cn/.4 C n/
3.
1
X
nD0
n
5
e
n
4.
1
X
nD0
p
n
n
2
C 1
5.
1
X
nD0
1:25
n
6.
1
X
nD0
e
n
Problem 4.49 Compute exactly:
1
X
nD1
1
n
2
C n
Problem 4.50 Compute exactly:
1
X
nD1
1
n
2
C 2n
Problem 4.51 Compute exactly:
1
X
nD1
1
n
2
C 5n
Problem 4.52
Demonstrate convergence of the series
1
X
nD1
sin
.n/
n
2
C n
Problem 4.53 Compute exactly:
1
X
nD0
1
4n
2
C 8n C 3
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