Which series should we compare this to?

Which way does the comparison go?

Describe the behavior of the series

We will use the DCT to determine if an infinite series converges or diverges.

In this section, we will determine whether a given series (with positive terms) converges or diverges by comparing it to a series whose behavior is known. Thus, it is important to recall the basic facts about -series and geometric series:

- -series: , converges if and diverges if
- geometric series: , converges if and diverges if

In the statement of the theorem below, the series is given and we are to determine whether it converges or diverges. To that end, we will compare the given series to a series of our choosing, denoted in the theorem by .

Direct Comparison Test Suppose for all .

- If converges, and for all , then converges.

In words: less than a convergent series is convergent.

- If diverges, and for all , then diverges.

In words: greater than a divergent series is divergent.

Unfortunately, the Direct Comparison Test is sometimes inconclusive, as noted in the following remark.

- If converges, and for all , then no conclusion can be made regarding

In words: greater than a convergent series is inconclusive.

- If diverges, and for all , then no conclusion can be made regarding

In words: less than a divergent series is inconclusive.

The qualifying statement **for all ** associated with the conditions and can be relaxed
to **for sufficiently large**.

Before we look at some examples, we review some basic facts about inequalities:

- (a)
- If then
- (b)
- If and then

example 1 Determine if the series converges or diverges.

We will use the DCT with the series This is a -series with and so it converges. To make the comparison, first note that for all . Taking reciprocals yields and so by the DCT (case 1), also converges.

We will use the DCT with the series This is a -series with and so it converges. To make the comparison, first note that for all . Taking reciprocals yields and so by the DCT (case 1), also converges.

(problem 1) Determine if the series converges or diverges:

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

example 2 Determine if the series converges or diverges.

We will use the DCT with the series This is a geometric series with common ratio, . Since , this series converges. To make the comparison, first note that for all . By taking the reciprocal of both sides, we have and multiplying by we get Therefore, by the DCT (case 1), also converges.

We will use the DCT with the series This is a geometric series with common ratio, . Since , this series converges. To make the comparison, first note that for all . By taking the reciprocal of both sides, we have and multiplying by we get Therefore, by the DCT (case 1), also converges.

(problem 2) Determine if the series converges or diverges:

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No Conclusion from DCT

example 3 Determine if the series converges or diverges.

We will use the DCT with the series This is a -series with and so it converges. To make the comparison, first note that for all . Dividing by , we have and so by the DCT (case 1), also converges.

We will use the DCT with the series This is a -series with and so it converges. To make the comparison, first note that for all . Dividing by , we have and so by the DCT (case 1), also converges.

(problem 3) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

example 4 Determine if the series converges or diverges.

We will use the DCT with the divergent harmonic series

We will use the DCT with the divergent harmonic series

To make the comparison, first note that for all . Dividing by , we have
Noting that this comparison goes in the wrong direction (smaller than a
divergent), the DCT gives **no conclusion** about the behavior of the series
.

(problem 4) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

example 5 Determine if the series converges or diverges.

We will use the DCT with the divergent harmonic series

We will use the DCT with the divergent harmonic series

To make the comparison, first note that for all . By taking the reciprocal of both sides, we have and so by the DCT (case 2), also diverges.

(problem 5) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

example 6 Determine if the series converges or diverges.

We will use the DCT with the divergent geometric series

We will use the DCT with the divergent geometric series

To make the comparison, first note that for all . Dividing by , we have and so by the DCT (case 2), also diverges.

(problem 6) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

example 7 Determine if the series converges or diverges.

We will use the DCT with the series which diverges (as shown in the section on the Integral Test). To make the comparison, first note that for all . Taking reciprocals, we have and so by the DCT (case 2), also diverges.

We will use the DCT with the series which diverges (as shown in the section on the Integral Test). To make the comparison, first note that for all . Taking reciprocals, we have and so by the DCT (case 2), also diverges.

example 8 Determine if the series converges or diverges.

We will use the DCT with the convergent p-series

We will use the DCT with the convergent p-series

To make the comparison, first note that for all . Dividing by , we have Noting that
this comparison goes in the wrong direction (larger than a convergent), the DCT
gives **no conclusion** about the behavior of the series . In the next section
on the Limit Comparison Test, we will learn a technique that allows us to
make a conclusion about this series by means of a slightly different type of
comparison.

(problem 8) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

Here is a detailed, lecture style video on the Direct Comparison Test:

_

(problem 9) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

(problem 10) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

(problem 11) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

(problem 12) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT

(problem 13) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

Which way does the comparison go?

for for

Describe the behavior of the series

Converges by DCT Diverges by DCT No
Conclusion from DCT