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8.1. SEQUENCES
Summary
In this section, we encountered the following important ideas:
• A sequence is a list of objects in a specified order. We will typically work with sequences
of real numbers and can also think of a sequence as a function from the positive
integers to the set of real numbers.
• A sequence {s n } of real numbers converges to a number L if we can make every value
of s k for k ≥ n as close as we want to L by choosing n sufficiently large.
• A sequence diverges if it does not converge.
Exercises
1. Finding limits of convergent sequences can be a challenge. However, there is a useful
tool we can adapt from our study of limits of continuous functions at infinity to use to
find limits of sequences. We illustrate in this exercise with the example of the sequence
ln(n)
n
.
(a) Calculate the first 10 terms of this sequence. Based on these calculations, do
you think the sequence converges or diverges? Why?
(b) For this sequence, there is a corresponding continuous function f defined by
f (x) =
ln(x)
x
.
Draw the graph of f (x) on the interval [0, 10] and then plot the entries of the
sequence on the graph. What conclusion do you think we can draw about the
sequence
ln(n)
n
if lim x→∞ f (x) = L? Explain.
(c) Note that f (x) has the indeterminate form
∞
∞ as x goes to infinity. What idea
from differential calculus can we use to calculate lim x→∞ f (x)? Use this method
to find lim x→∞ f (x). What, then, is lim n→∞
ln(n)
n ?
2. We return to the example begun in Preview Activity 8.1 to see how to derive the formula
for the amount of money in an account at a given time. We do this in a general setting.
Suppose you invest P dollars (called the principal) in an account paying r% interest
compounded monthly. In the first month you will receive
r
12 (here r is in decimal form;
e.g., if we have 8% interest, we write
0.08
12 ) of the principal P in interest, so you earn
P
r
12
8.1. SEQUENCES
Summary
In this section, we encountered the following important ideas:
• A sequence is a list of objects in a specified order. We will typically work with sequences
of real numbers and can also think of a sequence as a function from the positive
integers to the set of real numbers.
• A sequence {s n } of real numbers converges to a number L if we can make every value
of s k for k ≥ n as close as we want to L by choosing n sufficiently large.
• A sequence diverges if it does not converge.
Exercises
1. Finding limits of convergent sequences can be a challenge. However, there is a useful
tool we can adapt from our study of limits of continuous functions at infinity to use to
find limits of sequences. We illustrate in this exercise with the example of the sequence
ln(n)
n
.
(a) Calculate the first 10 terms of this sequence. Based on these calculations, do
you think the sequence converges or diverges? Why?
(b) For this sequence, there is a corresponding continuous function f defined by
f (x) =
ln(x)
x
.
Draw the graph of f (x) on the interval [0, 10] and then plot the entries of the
sequence on the graph. What conclusion do you think we can draw about the
sequence
ln(n)
n
if lim x→∞ f (x) = L? Explain.
(c) Note that f (x) has the indeterminate form
∞
∞ as x goes to infinity. What idea
from differential calculus can we use to calculate lim x→∞ f (x)? Use this method
to find lim x→∞ f (x). What, then, is lim n→∞
ln(n)
n ?
2. We return to the example begun in Preview Activity 8.1 to see how to derive the formula
for the amount of money in an account at a given time. We do this in a general setting.
Suppose you invest P dollars (called the principal) in an account paying r% interest
compounded monthly. In the first month you will receive
r
12 (here r is in decimal form;
e.g., if we have 8% interest, we write
0.08
12 ) of the principal P in interest, so you earn
P
r
12