3. PROP. 1. A series whose terms diminish in absolute value, and are, or end with becoming, alternately positive and negative, is convergent.

4

Let u1— u+u,—u+ &c. be the proposed series or its terminal portion, the part which it follows being in the latter case supposed finite. Then, writing it in the successive forms

and observing that u,u,, u,-u,, &c. are by hypothesis positive, we see that the sum of the series is greater than u-u, and less than u1. The series is therefore convergent.

u̟.

1

tends to a limit which is less than 1 and greater than *.

2

4. PROP. II. A series whose nth term is of the form u, sin ne (where 0 is not zero or an integral multiple of 2π) will converge if, for large values of n, un retains the same sign, continually diminishes as n increases, and ultimately vanishes.

Suppose u to retain its sign and to diminish continually as n increases after the term uɑ• Let

*

S=u2 sin ao+wa+ sin (a + 1) 0 + &c. .........

Ua

(3);

Although the above demonstration is quite rigorous, still such series present many analogies with divergent series and require careful treatment. For instance, in a convergent series where all the terms have the same sign, the order in which the terms are written does not affect the sum of the series. But in the given case, if we write the series thus,

in which form it is equally convergent, we find that its value lies between

5

4

and while that of the original series lies between 1

and 1

i.e.

3

2 3

between and 5.

1

2

Now Wa+1-UaUa+2-Ua+1' &c. are all negative, hence

S — u, cos ( a − 1 ) 0 < (Ua+1 − Ua) + (Va+s− Va+1) + &c.

2

numerically,

or <U∞ - Ua; :: <-ua, since u∞ = 0.

0

Hence the series is convergent unless sin be zero, i.e.

2

less be zero or an integral multiple of 2π.*

un

An exactly similar demonstration will prove the proposition for the case in which the nth term is u, sin (no – B).

is convergent unless be zero or a multiple of 2π. This is the case although, as we shall see, the series

5. The theory of the convergency and divergency of series whose terms are ultimately of one sign and at the same time converge to the limit 0, will occupy the remainder of this chapter and will be developed in the following order. 1st. A

* Malmstén (Grunert, vi. 38). A more general proposition is given by Chartier (Liouville, XVIII. 21).

fundamental proposition, due to Cauchy, which makes the test of convergency to consist in a process of integration, will be established. 2ndly. Certain direct consequences of that proposition relating to particular classes of series, including the geometrical, will be deduced. 3rdly. Upon those consequences, and upon a certain extension of the algebraical theory of degree which has been developed in the writings of Professor De Morgan and of M. Bertrand, a system of criteria general in application will be founded. It may be added that the first and most important of the criteria in question, to which indeed the others are properly supplemental, being founded upon the known properties of geometrical series, might be proved without the aid of Cauchy's proposition; but for the sake of unity it has been thought proper to exhibit the different parts of the system in their natural relation.

Fundamental Proposition.

6. PROP. III. If the function (x) be positive in sign but diminishing in value as a varies continuously from a to ∞o, then the series

(a)+(a + 1) + (a + 2) + &c. ad inf..........(4)

will be convergent or divergent according as [*$ (x) dx is

finite or infinite.

For, since (x) diminishes from x = a_ to x=a+1, and again from x = a + 1 to x = a + 2, &c., we have

x=

and so on, ad inf. Adding these inequations together, we

have

[®° $ (x) dæ < 4 (a) + $ (a + 1) + &c. ad inf. ..... (5).

a

and so on. Again adding, we have

[TM $ (x) dæ > $ (a + 1) + $ (a + 2) + &c. ...

α

(6).

Thus the integral («) da, being intermediate in value

between the two series

a

$ (a) + $ (a + 1) + $ (a + 2) + &c.

$ (a + 1) + $ (a + 2) + &c.

which differ by (a), will differ from the former series by a quantity less than (a), therefore by a finite quantity. Thus the series and the integral are finite or infinite together.

COR. If in the inequation (6) we change a into a compare the result with (5), it will appear that the series

$ (a) + $ (a + 1) + ☀ (a + 2) + &c. ad inf.

has for its inferior and superior limits

1, and

°4 (x) de, and f

(x) dx

dx........

a-1

(7).

7. The application of the above proposition will be sufficiently explained in the two following examples relating to geometrical series and to the other classes of series involved in the demonstration of the final system of criteria referred to in Art. 5.

Ex. 1. The geometrical series

1+h+h2+h3+ &c. ad inf.

is convergent if h<1, divergent if h 1.

>

The general term is h, the value of x in the first term being 0, so that the test of convergency is simply whether hdx is infinite or not. Now

If h>1 this expression becomes infinite with x and the series is divergent. If h<1 the expression assumes the finite The series is therefore convergent.

value

- 1

log h

If h1 the expression becomes indeterminate, but, proceeding in the usual way, assumes the limiting form xh which becomes infinite with x. Here then the series is divergent.

+ &c.

a (loga)" (a+1) {log (a+1)}"

aloga (logʻloga)TM* (a+1)log(a+1){loglog(a+1)}m

being positive, are convergent if m>1, and divergent

* The convergency of these series can be investigated without the use of the Integral Calculus. See Todhunter's Algebra (Miscellaneous Theorems), or Malmstén (Grunert, vIII. 419).