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8.3. SERIES OF REAL NUMBERS
8.3 Series of Real Numbers
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is an infinite series?
• What is the nth partial sum of an infinite series?
• How do we add up an infinite number of numbers? In other words, what does it
mean for an infinite series of real numbers to converge?
• What does is mean for an infinite series of real numbers to diverge?
Introduction
In Section 8.2, we encountered several situations where we naturally considered an infinite
sum of numbers called a geometric series. For example, by writing
N = 0.1212121212 · · · =
12
100
+
12
100
·
1
100
+
12
100
·
1
100 2 + · · ·
as a geometric series, we found a way to write the repeating decimal expansion of N
as a single fraction: N =
4
33 . There are many other situations in mathematics where
infinite sums of numbers arise, but often these are not geometric. In this section, we begin
exploring these other types of infinite sums. Preview Activity 8.3 provides a context in
which we see how one such sum is related to the famous number e.
Preview Activity 8.3. Have you ever wondered how your calculator can produce a
numeric approximation for complicated numbers like e, π or ln(2)? After all, the only
operations a calculator can really perform are addition, subtraction, multiplication, and
division, the operations that make up polynomials. This activity provides the first steps in
understanding how this process works. Throughout the activity, let f (x) = e x .
(a) Find the tangent line to f at x = 0 and use this linearization to approximate
e. That is, find a formula L(x) for the tangent line, and compute L(1), since
L(1) ≈ f (1) = e.
(b) The linearization of e x does not provide a good approximation to e since 1 is
not very close to 0. To obtain a better approximation, we alter our approach
a bit. Instead of using a straight line to approximate e, we put an appropriate
bend in our estimating function to make it better fit the graph of e x for x close
to 0. With the linearization, we had both f (x) and f ′ (x) share the same value
8.3. SERIES OF REAL NUMBERS
8.3 Series of Real Numbers
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is an infinite series?
• What is the nth partial sum of an infinite series?
• How do we add up an infinite number of numbers? In other words, what does it
mean for an infinite series of real numbers to converge?
• What does is mean for an infinite series of real numbers to diverge?
Introduction
In Section 8.2, we encountered several situations where we naturally considered an infinite
sum of numbers called a geometric series. For example, by writing
N = 0.1212121212 · · · =
12
100
+
12
100
·
1
100
+
12
100
·
1
100 2 + · · ·
as a geometric series, we found a way to write the repeating decimal expansion of N
as a single fraction: N =
4
33 . There are many other situations in mathematics where
infinite sums of numbers arise, but often these are not geometric. In this section, we begin
exploring these other types of infinite sums. Preview Activity 8.3 provides a context in
which we see how one such sum is related to the famous number e.
Preview Activity 8.3. Have you ever wondered how your calculator can produce a
numeric approximation for complicated numbers like e, π or ln(2)? After all, the only
operations a calculator can really perform are addition, subtraction, multiplication, and
division, the operations that make up polynomials. This activity provides the first steps in
understanding how this process works. Throughout the activity, let f (x) = e x .
(a) Find the tangent line to f at x = 0 and use this linearization to approximate
e. That is, find a formula L(x) for the tangent line, and compute L(1), since
L(1) ≈ f (1) = e.
(b) The linearization of e x does not provide a good approximation to e since 1 is
not very close to 0. To obtain a better approximation, we alter our approach
a bit. Instead of using a straight line to approximate e, we put an appropriate
bend in our estimating function to make it better fit the graph of e x for x close
to 0. With the linearization, we had both f (x) and f ′ (x) share the same value