In most series of positive terms required in applications the terms tend to zero, steadily diminishing. To such series we can apply a test called Cauchy' s condensation test, viz., that Z4 (n) is convergent, if a"4(an) is convergent, a being any positive integer. We have also Maclaurin's test, viz., that if =Ca), and so en, are each convergent if p> I but divergent if p i.
A series of tests, called the ratio-tests, applicable only to series in which
is of great historical as well as of great practical interest. The more elaborate of them are asso ciated with the names of A. De Morgan, A. Cauchy, J. Bertrand and others and can be proved by means of the comparison test and the convergence or divergence of the series just mentioned. They are most readily established by means of a general theorem associated with the names of Dini and Kummer, which is as follows: If a function f(n) can be found such that or, if not tending to a limit, is always greater than some fixed positive number, the series
is convergent. If, moreover, If
is divergent, then, if
- f (n 1)--> 1 < o, or un+1 is never positive,
is divergent. By taking f(n) in succes sion as 1, n, n log n,• • • we obtain the following series of tests: (I) If
I, or
is always greater than some fixed number > 1,
is convergent. If
(2) If n{
- then Zu„ is convergent if 1> 1,
gent if 1
(3) If - I) - I log n-*l, then is convergent if 1>i, divergent if l< 1, and nothing is settled if 1= 1, with the same modifications as before.
The following rule will cover most cases. If can be expressed in the form X> i, then /u„ is con vergent if /I> 1, divergent if 1.1 I.
Alternative forms of the second and third tests are as follows: If n or log n log nlog(u„/u„.“)}-->l, the series is convergent, if 1> 1, divergent if l< 1, and so on. There are still other forms of these tests, superficially different but essentially the same.
Another test, known as Cauchy's test, of great theoretical importance, is the following. If
stage,
is never greater than 1, some fixed number < I,
is convergent. If u„'In->l> 1, or if after any stage, there are an unlimited number of integers n for which
is as great as unity, the series is divergent. For under these conditions u„ cannot tend to zero. This test is applicable to series in which
does not tend steadily to zero, as for instance the series Zan
where a< I. For
though it tends to no definite limit, cannot exceed a. It may be remarked that, if
has a limit,
has the same limit, though the converse is not necessarily true.
If the terms of a convergent series of positive terms are de ranged according to any law to form a new series, the new series is also convergent and its sum is that of the original series. If is convergent so is where is any positive number less than some fixed number independent of n. Similar results hold for a divergent series of positive terms. It is generally possible from these considerations to determine the convergence or divergence of a series of positive terms, in which does not tend steadily to zero.
Series Whose Terms Are of Variable Sign. Absolute Con vergence.—When a series consists of terms not all of the same sign, it may happen that the series I lu„I is convergent. In this case is convergent and is said to be absolutely convergent. The positive terms alone form a convergent series and the negative terms alone form another convergent series. If the terms of such a series are deranged to form a second series, the second series is convergent and its sum is the same as that of the original series. If is divergent, the series is said to be non absolutely or conditionally convergent. The positive terms alone form a divergent series and the negative terms alone form a second divergent series. Its sum depends upon the order of its terms. It has been proved by Riemann that the terms of such a series can be arranged to form a series which shall converge to any assigned sum or even to diverge. The series + • • • is convergent, its sum being log 2. The series I-F-1-1+*-4-1+, obtained by rearranging its terms, is convergent but its sum is obtained by rearranging its terms, is divergent.