8.3. SERIES OF REAL NUMBERS
481
⊳
We can generalize the argument in Activity 8.14 in the following way. Consider the
series
a k . If
a k+1
a k
≈ r
for large values of k, then a k+1 ≈ ra k for large k and the series
a k is approximately the
geometric series
ar k for large k. Since the geometric series with ratio r converges only
for −1 < r < 1, we see that the series
a k will converge if
lim
k→∞
a k+1
a k
= r
for a value of r such that |r | < 1. This result is known as the Ratio Test.
The Ratio Test. Let
a k be an infinite series. Suppose
lim
k→∞
|a k+1 |
|a k |
= r.
1. If 0 ≤ r < 1, then the series
a k converges.
2. If 1 < r, then the series
a k diverges.
3. If r = 1, then the test is inconclusive.
Note well: The Ratio Test takes a given series and looks at the limit of the ratio of
consecutive terms; in so doing, the test is essentially asking, “is this series approximately
geometric?” If the series can be thought of as essentially geometric, the test use the limiting
common ratio to determine if the given series converges.
We have now encountered several tests for determining convergence or divergence of
series. The Divergence Test can be used to show that a series diverges, but never to prove
that a series converges. We used the Integral Test to determine the convergence status of
an entire class of series, the p-series. The Limit Comparison Test works well for series that
involve rational functions and which can therefore by compared to p-series. Finally, the
Ratio Test allows us to compare our series to a geometric series; it is particularly useful
for series that involve nth powers and factorials. Two other tests, the Direct Comparison
Test and the Root Test, are discussed in the exercises. Now it is time for some practice.
Activity 8.15.
Determine whether each of the following series converges or diverges. Explicitly state
which test you use.
(a)
k
2 k
481
⊳
We can generalize the argument in Activity 8.14 in the following way. Consider the
series
a k . If
a k+1
a k
≈ r
for large values of k, then a k+1 ≈ ra k for large k and the series
a k is approximately the
geometric series
ar k for large k. Since the geometric series with ratio r converges only
for −1 < r < 1, we see that the series
a k will converge if
lim
k→∞
a k+1
a k
= r
for a value of r such that |r | < 1. This result is known as the Ratio Test.
The Ratio Test. Let
a k be an infinite series. Suppose
lim
k→∞
|a k+1 |
|a k |
= r.
1. If 0 ≤ r < 1, then the series
a k converges.
2. If 1 < r, then the series
a k diverges.
3. If r = 1, then the test is inconclusive.
Note well: The Ratio Test takes a given series and looks at the limit of the ratio of
consecutive terms; in so doing, the test is essentially asking, “is this series approximately
geometric?” If the series can be thought of as essentially geometric, the test use the limiting
common ratio to determine if the given series converges.
We have now encountered several tests for determining convergence or divergence of
series. The Divergence Test can be used to show that a series diverges, but never to prove
that a series converges. We used the Integral Test to determine the convergence status of
an entire class of series, the p-series. The Limit Comparison Test works well for series that
involve rational functions and which can therefore by compared to p-series. Finally, the
Ratio Test allows us to compare our series to a geometric series; it is particularly useful
for series that involve nth powers and factorials. Two other tests, the Direct Comparison
Test and the Root Test, are discussed in the exercises. Now it is time for some practice.
Activity 8.15.
Determine whether each of the following series converges or diverges. Explicitly state
which test you use.
(a)
k
2 k