8.3. SERIES OF REAL NUMBERS
473
Since a 1 + a 2 + · · · + a m is a finite number, the series
∞
k=1 a k will converge if and only
if the series
∞
k=m+1 a k converges. Because the starting index of the series doesn’t affect
whether the series converges or diverges, we will often just write
a k
when we are interested in questions of convergence/divergence and not necessarily the
exact sum of a series.
In Section 8.2 we encountered the special family of infinite geometric series whose
convergence or divergence we completely determined. Recall that a geometric series is a
special series of the form
∞
k=0 ar k where a and r are real numbers (and r 1). We found
that the nth partial sum S n of a geometric series is given by the convenient formula
S n =
1 − r n
1 − r
,
and thus a geometric series converges if |r | < 1. Geometric series diverge for all other
values of r. While we have completely determined the convergence or divergence of
geometric series, it is generally a difficult question to determine if a given nongeometric
series converges or diverges. There are several tests we can use that we will consider in
the following sections.
The Divergence Test
The first question we ask about any infinite series is usually “Does the series converge or
diverge?” There is a straightforward way to check that certain series diverge; we explore
this test in the next activity.
Activity 8.8.
If the series
a k converges, then an important result necessarily follows regarding the
sequence {a n }. This activity explores this result.
Assume that the series
∞
k=1 a k converges and has sum equal to L.
(a) What is the nth partial sum S n of the series
∞
k=1 a k ?
(b) What is the (n − 1)st partial sum S n−1 of the series
∞
k=1 a k ?
(c) What is the difference between the nth partial sum and the (n − 1)st partial
sum of the series
∞
k=1 a k ?
(d) Since we are assuming that
∞
k=1 a k = L, what does that tell us about
lim n→∞ S n ? Why? What does that tell us about lim n→∞ S n−1 ? Why?
(e) Combine the results of the previous two parts of this activity to determine
lim n→∞ a n = lim n→∞ (S n − S n−1 ).
473
Since a 1 + a 2 + · · · + a m is a finite number, the series
∞
k=1 a k will converge if and only
if the series
∞
k=m+1 a k converges. Because the starting index of the series doesn’t affect
whether the series converges or diverges, we will often just write
a k
when we are interested in questions of convergence/divergence and not necessarily the
exact sum of a series.
In Section 8.2 we encountered the special family of infinite geometric series whose
convergence or divergence we completely determined. Recall that a geometric series is a
special series of the form
∞
k=0 ar k where a and r are real numbers (and r 1). We found
that the nth partial sum S n of a geometric series is given by the convenient formula
S n =
1 − r n
1 − r
,
and thus a geometric series converges if |r | < 1. Geometric series diverge for all other
values of r. While we have completely determined the convergence or divergence of
geometric series, it is generally a difficult question to determine if a given nongeometric
series converges or diverges. There are several tests we can use that we will consider in
the following sections.
The Divergence Test
The first question we ask about any infinite series is usually “Does the series converge or
diverge?” There is a straightforward way to check that certain series diverge; we explore
this test in the next activity.
Activity 8.8.
If the series
a k converges, then an important result necessarily follows regarding the
sequence {a n }. This activity explores this result.
Assume that the series
∞
k=1 a k converges and has sum equal to L.
(a) What is the nth partial sum S n of the series
∞
k=1 a k ?
(b) What is the (n − 1)st partial sum S n−1 of the series
∞
k=1 a k ?
(c) What is the difference between the nth partial sum and the (n − 1)st partial
sum of the series
∞
k=1 a k ?
(d) Since we are assuming that
∞
k=1 a k = L, what does that tell us about
lim n→∞ S n ? Why? What does that tell us about lim n→∞ S n−1 ? Why?
(e) Combine the results of the previous two parts of this activity to determine
lim n→∞ a n = lim n→∞ (S n − S n−1 ).