+

(1 − x)3 − (1 − x)3 (1-x)2 1-x

Hence the general term of the series

1+3x+7x2 + ... is xn-1 {n (n + 1) − 2n+1} = (n2 − n + 1) xn−1.

Thus the general term of the given series is n2 - n+1.

Having found the general term of the series the sum of the first n terms can be written down, for the sum of n terms of the series

whose nth term is n (n − 1) + 1

1

is — (n − 1) n (n + 1) + n.

3

Ex. 3. Find the nth term of the series 2+2+8+20+....

Considered as a recurring series of the lowest possible order, the generating function of 2+2x+8x2 + 20x3 + ... will be found to be

2-2x 1-2x-2.x2

Now the factors of 1-2x - 2x2 are irrational, and therefore the nth term of the series, considered as a recurring series of the second order, will be a complicated expression containing radicals.

On the other hand, by the method of Art. 325, we should be led to conclude that the nth term of the series was (3n2 - 9n+8) xn−1, which by Art. 330 is a recurring series of the third order.

As we have already remarked, the actual law of a series cannot be determined from any finite number of its terms, and the above is a case in which it would be difficult to decide as to what is the simplest law that the few terms given obey, for the recurring series of the lowest order which has the given terms for its first four terms is not the recurring series which gives the simplest expression for the nth term.

CONVERGENCY AND DIVERGENCY.

332. We shall now investigate certain theorems in convergency which were not considered in Chapter XXI.

333. Convergency of infinite products. A product composed of an infinite number of factors cannot be convergent unless the factors tend to unity as their limit; for otherwise the addition of a factor would always make a finite change in the product of the preceding factors, and there could be no definite quantity to which the product approached without limit as the number of factors was indefinitely increased.

It is therefore only necessary to consider infinite products of the form

where

II (1 + u„) = (1 + u1) (1 + u2) (1 + u ̧).......(1+u„).......,

Ип

becomes indefinitely small as n is indefinitely increased; and the convergency or divergency of such products is determined by the following

Theorem. The infinite product II (1+u,), in which all the factors are greater than unity, is convergent or divergent according as the infinite series Zu, is convergent or divergent.

Since e* > 1 + x, for all positive values of x, it follows that

(1 + u ̧) (1 + u2) (1 + U2).....< eÛ1. eÛ2. €Û3..... < eÛ2+u2+uz+ Hence, if Σu, be convergent, II (1+u,) will also be convergent.

Again,

(1 + u1) (1 + u2) >1+u,

+ Иза

(1 + u1) (1 + u2) (1 + u ̧) > 1 + u ̧ +U2+U ̧›

and so on, so that

II (1 + u„) > 1 + Σu„.

Hence, if Σu, be divergent, II (1+u) will also be divergent.

a (a+1)(a+2)...(a + n − 1)
b (b+1) (b+2)...(b+ n − 1)

is infinite or zero, when

Ex. 1. To shew that n is indefinitely increased, according as a is greater or less than b.

S. A.

27

For, if a > b, the expression may be written in the form

+.. b+1 b+2

is therefore infinite when n is infinite, a being greater than b.

is a divergent series: the given expression

a

Hence the given series is convergent, and its sum is then

b-a-x

if b> a+x. Also the series is divergent if b<a+x.

334. The Binomial Series. We have already proved that the binomial series, namely

is convergent or divergent, for all values of m, according as x is numerically less or greater than unity.

n

Now we know that the terms of this series are alternately positive and negative after the rth term, where r is the first positive integer greater than m+1. Moreover the ratio u+1/u, is numerically less or greater than unity according as m+1 is positive or negative. The series will therefore, from theorem V. Chapter XXI. be convergent when m+1 is positive provided the nth term decreases without limit as n is increased without limit.

Now, if m+1 be positive and less than r, the product of the factors from the rth onwards is greater than

and the product of the preceding factors is finite.

Hence, when n is increased without limit, 1/u, is infinitely great, and therefore u, indefinitely small, provided 1+m be positive.

Thus the binomial series is convergent if x = 1, provided m-1.

If a

x=- - 1, the series becomes

The sum of n terms of the above series is easily found to be [see Art. 283 or Art. 321]

(1 — m) (2 — m) (3 m)... (n— 1 — m)

1.2.3...(n-1)

The sum of n terms of the series is therefore [Ex. 1, Art. 333], zero or infinite, when n is infinite, according as m is positive or negative.

Thus the binomial series is convergent when x = - 1, provided m is positive.

+

...

335. Cauchy's Theorem. If the series u1 + U2 + U2 +un+... have all its terms positive, and if each term be less than the preceding, then the series will be convergent or divergent according as the series u1 + au1 + a3ua2 + ..... +a"uan+... is convergent or divergent, a being any positive integer.

For, since each term is less than the preceding, we have the following series of relations

Ua + 1 + Ua + 2 ••• Ua2 < (a2 − a) u, < (a − 1) au

Ua” +1+Ua” +2 + ... + Uq"+1< (a"+1 — a") Ua" < (a − 1) a"uɑ".

Hence, by addition, S<(a-1) + u1........

(I),

where S and Σ stand for the sum of the first and second series respectively.

a (Uan-1+1+Uan-1+2+...+Ua") > a (a" — a"-1) ua"> a"uq".

.(II).

From I and II it follows that if S is finite so also is Σ, and that if S is infinite so also is Σ.

1

n (log n)k

is convergent if k be greater

Ex. To shew that the series than unity, and divergent if k be equal or less than unity. By Cauchy's theorem the series will be convergent or divergent

an

according as the series whose general term is
or divergent.

is convergent

a" (log an)