The series introduced thus far have been those with non-negative terms. In this section, series with mixed signs such as the alternating series will be introduced. Several convergence testing methods, such as the alternating series test, ratio test and root test, will be presented.

As the name of the alternate series imply, these series have consecutive terms with alternate signs. An example of alternating series is

The above series is also known as the alternating harmonic series. Note that the last term of the series has the form a_n = (-1)^n-1b_n where b_n is positive. The plot of the partial sum as a function of n is shown in the figure on the left.

Plot of the Partial Sum for an
Alternate Series Σ[(-1)ⁿ/2ⁿ] as a
Function of n.

According to the alternating series test, when a series consists of
(a) terms with alternate signs,
(b) terms which decreases in magnitude
     (i.e., for all n), and
(c) the n^th term (a_n) has limit of zero
     (i.e., ),
then the series converges.

Take the series
for example. Obviously, this series has alternating terms. The absolute magnitude of the terms decrease as n increases (i.e., ). And as n tends to infinity, n^th term has a limit of zero. Hence, by alternating series test, this series converges.

Often partial sum S_n is used as an estimation to the total sum S of a convergent series. How many terms are required to provide a reasonable good estimation? What is the error involved? For a series which satisfies the alternating series test, the remainder of using S_n as an estimation of S is given by

Plot of the Partial Sum for an
Alternate Series Σ[(-1)ⁿ⁺¹/n²] as a
Function of n.

A series Σa_n is said to converge absolutely if the series of absolute terms () converges. A series is convergent if it is absolute convergent. However, a series is conditionally convergent if it is convergent but not absolutely convergent.

For example, the series

is absolute convergent since it was proved previously, using the basic comparison and integral tests, that the following series (p-series with p = 2)

converges.

On the other hand, the alternating harmonic series

     
only converges conditionally. The above series is convergent but the series with absolute terms

diverges.

Another useful test in examining whether a series is convergent or divergent is the ratio test, which is stated as follows:

(1) The seriesis absolutely convergent if
     

(2) The series is divergent if
     

Note that if the limit tends to 1 then the ratio test gives no information regarding the convergency. The ratio test is particularly useful when examining series with terms that consist of factorials and n^th powers of constants.

Take the series    for example. According to the ratio test:

Hence, this series is absolutely convergent.

Besides the ratio test, the root test can also be used to examine if a series is convergent or divergent. The root test is convenient in determining series with terms raised to the n^th power, and it is stated as follows:

(1) The seriesis absolutely convergent if

(2) The series is divergent if

The series was shown to be absolute convergent using the ratio test. Using the root test:

it is also determined that the series is absolute convergent.