**11.6** Absolute Convergence and the

### Ratio and Root Tests

### Absolute Convergence and the Ratio and Root Tests

Given any series Σ a* _{n}*, we can consider the corresponding
series

*| a*_{n}*| = | a*_{1}*| + |a*_{2}*| + | a*_{3}| + ^{. . . }

whose terms are the absolute values of the terms of the original series.

### Absolute Convergence and the Ratio and Root Tests

Notice that if Σ a* _{n}* is a series with positive terms, then

*| a*_{n}*| = a** _{n}* and so absolute convergence is the same as
convergence in this case.

### Example 1

The series

is absolutely convergent because

*is a convergent p-series (p = 2).*

### Example 2

We know that the alternating harmonic series

is convergent, but it is not absolutely convergent because the corresponding series of absolute values is

*which is the harmonic series (p-series with p = 1) and is*
therefore divergent.

### Absolute Convergence and the Ratio and Root Tests

Example 2 shows that the alternating harmonic series is conditionally convergent. Thus it is possible for a series to be convergent but not absolutely convergent.

However, the next theorem shows that absolute convergence implies convergence.

### Example 3

Determine whether the series

is convergent or divergent.

Solution:

This series has both positive and negative terms, but it is not alternating. (The first term is positive, the next three are negative, and the following three are positive: The signs

change irregularly.)

*Example 3 – Solution*

We can apply the Comparison Test to the series of absolute values

*Since | cos n | ≤ 1 for all n, we have*

We know that Σ 1/n^{2} *is convergent (p-series with p = 2) and*
therefore Σ |cos n|/n^{2} is convergent by the Comparison

Test.

cont’d

*Example 3 – Solution*

Thus the given series Σ (cos n)/n^{2} is absolutely convergent
and therefore convergent by Theorem 3.

cont’d

### Absolute Convergence and the Ratio and Root Tests

The following test is very useful in determining whether a given series is absolutely convergent.

### Absolute Convergence and the Ratio and Root Tests

**Note:**

Part (iii) of the Ratio Test says that if

the test gives no information. For instance, for the
convergent series Σ 1/n^{2} we have

### Absolute Convergence and the Ratio and Root Tests

whereas for the divergent series Σ 1/n we have

Therefore, if the series Σ a* _{n}* might
converge or it might diverge. In this case the Ratio Test
fails and we must use some other test.

### Example 5

Test the convergence of the series Solution:

*Since the terms a*_{n}*= n*^{n}*/n! are positive, we don’t need the *
absolute value signs.

*Example 5 – Solution*

*as n*

*Since e > 1, the given series is divergent by the Ratio Test.*

cont’d

### Absolute Convergence and the Ratio and Root Tests

**Note: **

Although the Ratio Test works in Example 5, an easier method is to use the Test for Divergence. Since

*it follows that a*_{n}*does not approach 0 as n* . Therefore
the given series is divergent by the Test for Divergence.

### Absolute Convergence and the Ratio and Root Tests

*The following test is convenient to apply when n th powers *
occur.

### Absolute Convergence and the Ratio and Root Tests

If then part (iii) of the Root Test says that
the test gives no information. The series Σ a* _{n}* could

converge or diverge.

*(If L = 1 in the Ratio Test, don’t try the Root Test because L*
*will again be 1. And if L = 1 in the Root Test, don’t try the *
Ratio Test because it will fail too.)

### Example 6

Test the convergence of the series Solution:

Thus the given series is absolutely convergent (and therefore convergent) by the Root Test.

### Rearrangements

### Rearrangements

The question of whether a given convergent series is absolutely convergent or conditionally convergent has a bearing on the question of whether infinite sums behave like finite sums.

If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. But this is not always the case for an infinite series.

### Rearrangements

**By a rearrangement of an infinite series Σ a*** _{n}* we mean a
series obtained by simply changing the order of the terms.

For instance, a rearrangement of Σ a* _{n}* could start as follows:

*a*_{1} *+ a*_{2} *+ a*_{5} *+ a*_{3} *+ a*_{4} *+ a*_{15} *+ a*_{6} *+ a*_{7} *+ a*_{20} + ^{. . . }

It turns out that

if Σ a_{n}*is absolutely convergent series with sum s,*

then any rearrangement of Σ a_{n}*has the same sum s.*

### Rearrangements

However, any conditionally convergent series can be rearranged to give a different sum. To illustrate this fact let’s consider the alternating harmonic series

If we multiply this series by we get

### Rearrangements

Inserting zeros between the terms of this series, we have

Now we add the series in Equations 6 and 7:

### Rearrangements

Notice that the series in (8) contains the same terms as in (6) but rearranged so that one negative term occurs after each pair of positive terms. The sums of these series,

however, are different. In fact, Riemann proved that

if Σ a_{n}*is a conditionally convergent series and r is any *
real number whatsoever, then there is a rearrangement
of Σ a_{n}*that has a sum equal to r.*