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since p is less than 1. And this quantity is finite; consequently the given series is convergent. Its terms evidently become less and less, because they are less than the corresponding terms of a decreasing geometrical series.

If r is greater than 1, let p be greater than 1 and less than r; then for some determinate value of x, and for all other greater values,

Uz+1 > puz, Uz+> pUz+1) Uz+3 > p U2+2...... ; consequently,

Uz+uz+1+U2+2 + ...... > Uz{1+p+pa + ......}; the right-hand member of which = 00, when the number of the terms is infinite; thus the sum of the series is infinite, and the series is accordingly divergent. If r = 1, we can make no inference; and in this case the series may be either convergent or divergent.

If the series is arranged in ascending powers of a variable t, say, and is such that u, = 1,12; then 4+1 = 4x+1 t; and if

и, А. *2+1 = r, when x = oo, the discriminating quantity = rt; so that the series is convergent or divergent accordingly as t is less than or greater than r-1. If t=p-1, we can affirm nothing as to convergence or divergence. Ex. 1. Let the series be 1 + + + + ...; then

Uz+1 = = 0, when x = 0;

Uz X thus, whatever is the value of t, the series is convergent. I may however observe, that if t is greater than 1, the series begins divergently, and begins to converge at that term for which is less than l; that is, when x is greater than t.

The test of convergence and divergence given in this Article is of all perhaps the most easy of application, and is accordingly that which should be at once employed. And it is only when the ratio of uz+1 to uz is equal to 1, that the test fails, and we are obliged to have recourse to other criteria.

182.] If f(x) is a function of x, positive in sign, and decreasing in value as x increases from x = a to x = 0o, and ultimately equal to 0, when x = 0o, then the series Za f(x), that is, f(a)+f (a +1)+f(a+2)+...

(13) PRICE, VOL. II.


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will be convergent or divergent according as or infinite.

By the theory of definite integration

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and applying the theorem given in (228), Art. 116, where 0 is the general symbol of a positive proper fraction, [ f(x) dx =f(a+0) / **dx+f(a+1+0)/a*dx +...,

= f(a+0) + f(a+1+0) + f(a+2+0) +.... (15) Now as 0 is a positive proper fraction, 1 and 0 are its limits; we may consequently deduce from this equation two inequalities corresponding to these values of 0; and as f (x) is positive, and decreases as x increases, when 0 = 0, the second member of (15) is greater than the first; and when 0 = 1, the second member is less than the first; hence

f(a)+f(a+1)+f(a+2) + ...... > f(x) dr ; (16)


f(a+1)+f(a+2) +..... < / f(x) dx. (17) Then as the sum of the given series is greater than / f(x) dx, and is less than f(x) dx + f(a), where f(a) is a finite quantity, it is evident that the sum is infinite or finite, that is, the series is divergent or convergent, according as / f(x) dx is infinite or finite. This quantity I shall call the discriminating quantity.

This is the most general criterion of the convergence and divergence of series which has as yet been discovered. The difficulty of its application consists in the integration. By a farther inquiry however into the general theory of series, we shall be led to a classification of them as to order and degree, and the preceding theorem will supply certain derivative tests convenient for the purpose.

The investigation also gives narrow limits within which the sum of a series of the form (13) is contained.

In (17) let a be replaced by a-1; then

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And if the sum of the series = s, from this and (16) it follows that

lo f(x) dx > s > f(x) dx ; (19) so that s is contained within these two definite integrals.

183.] The following are examples in which the preceding test is applied.

Ex. 1. The series in geometrical progression a, ar, ara, ... , where r is a positive quantity.

Taking the general term of the series to be art, the successive terms of the series will be formed by putting x = 0, 1, 2, 3, ... ; so that the discriminating quantity is art dx. Now

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= oo, if r is not less than 1. Hence a series in geometrical progression is convergent or divergent according as r is less than, or not less than, 1. Ex. 2. Let the series be 1 1

1 ant Toilm+ a2m +...; which is Lor". (20) In this example the discriminating quantity is -mdr. Now

00-m+1-a-m+1 | x-mdx = =

- m +1

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1, if m is greater than 1;

= oo, if m is not greater than 1. Hence the series is convergent or divergent according as m is greater, or is not greater, than 1. If m= 1, the series is harmonic, and we have hereby another proof that such a series is divergent.

Ex. 3. La x-1 (log x)-m is convergent or divergent according as m is greater than, or not greater than, 1. Here the discriminating condition is /

condition is dx

Ja 2 (log x):

pool dx (log oo)-m+1-(log a)–m+1 Ja x(log x)"

-m +1 (log a)-m+1

-, if m is greater than 1;

= co, if m is not greater than 1. Ex. 4. La {x log x (log log x)"}-1 is convergent or divergent, according as m is greater, or is not greater, than 1.

Ex. 5. Generally if log log log ... (to n symbols) ... x is denoted by log" X, then a {x log x loge x ... login-1 x (log" x)"}-1 is convergent or divergent, according as m is greater than, or not greater than, 1.

Ex. 6. Zo quis convergent.

Ex. 7. E log is divergent. Hence 2+ 3+ 4+... nå = , when n = 0.

Ex. 8. Ei login) * is convergent.


184.] Although the criterion of the preceding article is theoretically always sufficient, yet it is in many cases inapplicable, because the definite integral which gives the discriminating quantity cannot be found. We can however derive from it certain other criteria, which are frequently easy of application; this is effected by a comparison of the given series, with other series which are known to be convergent or divergent. Let there be two series v = Uz + Uz +Uz +...

(21) V = V1 + V2 + V3+...; the ratio of the nth terms of which is equal to a finite quantity k, when n = 00: so that Un = k vn, when n = oc; then these two series are either both convergent or both divergent.

Let us commence with the nth terms respectively of the two series; and let un = km Vn, Un+1 = kn+1 Vn+1, Un+2 = kn+2 Vn+2, ...; then since k, = kn+1 = kn+2 = ... = k, when n=00; so when n is very great, the difference between k and each of these quantities is an infinitesimal; and we shall have generally k, = k+i, where i is an infinitesimal; and ultimately when n = 00, i = 0, and

Un + Un+1 +Wn+2+... = k{vn+ vn+1 + Vn+2+...}. (23)

Hence it appears that if either of these sums is a finite quantity, the other is also finite; and consequently the series are either both convergent or both divergent.

If series of the form (21) and (22) are so related to each other, that when n = oo, Un = kvm, k being a finite quantity, they are called comparable. If however k = 0 or = 0, they are said to be incomparable. Ifk = 0, (21) is incomparably higher than (22). If k = 0, (22) is incomparably higher than (21).

Hence it is evident that if a series is convergent, all comparable series and all incomparably lower series are also convergent. If a series is divergent, all comparable series and also incomparably higher series are also divergent.

185.] I proceed now to apply this theory; and I shall take for the series of comparison those of which the character as to convergence or divergence has been demonstrated in Art. 183.

Let the series, the character of which is to be determined, be

otto, in which f (x) = 0, when x = 0: and let us in the first place compare it with a x-m, of which the character is determined in Ex. 2 of the preceding Article. Let us take m to be such that these series are comparable; then if k is the ratio of the corresponding terms when x = on,

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=a finite quantity, suppose, when x = 00: so that f'(x) and 2m-1 are infinities of the same order ;

k = xf'(x) = k *f'(x);

mf (x)' :: m = SOCIO), when « = . (24) and as Xox-m is convergent when m is greater than 1, and divergent when m is less than 1; so will the given series

& the be convergent if ICT) , when x = co, is greater than 1; and divergent if y(0), when x = oo, is less than 1. Hence

f(x) the discriminating quantity of

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