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DEF. 2. An endless series is said to be Divergent if the sum of any number of its terms may be made greater than any assignable finite quantity by sufficiently increasing the number of terms.

441. According to the above definitions there are two kinds of series which are neither convergent nor divergent.

Thus a series which is such that, when n= f (r) the sum tends towards some limit A when r is indefinitely increased, and when n=0 (r) the sum tends towards some other limit B when r is indefinitely increased, and so on : then the series is neither convergent nor divergent. For example,

a - a + a a + ...... = 0 or a, according as n = 2r or 2r + 1.

1-1 + 2 2 + 3 3 + according as n = 2r or 2r + 1.

If all the different limits are finite, the series may be called Indeterminately finite: if one of the limits is infinite, the series may be called Indeterminately infinite.

442. The above distinctions may be expressed by the use of the term Limit; thus

An endless series is convergent, if the sum of n terms, when n=co, has some single finite limit.

It is divergent, if, in the limit, its sum = when n=0.

It is indeterminately finite, if the sum of n terms, when n=c, has more than one limit which is always finite.

It is indeterminately infinite, if the sum of n terms, when n=0, has infinite and finite limits.

443. Since the nth term of a series is equal to the difference between the sum of n terms and of n-1 terms, a series cannot be convergent unless, when n is infinite, the limit of the nth term is

This is a necessary but not a sufficient condition for convergency

a

zero.

If the limit of the nth term is not zero, but finite, the sum of the series may be finite for all values of n however great, but it cannot have a single limit. Thus the series

2n-1 -1+ - +1 -... +(-1)"

+

n

is finite, but it has a different limit according as n is even or odd.

If the limit of the nth term is infinite, the sum of the series may be finite for some values of n however great, but it cannot be finite for all values of n. Thus the series

2m2 7

1 2n? - 1 -0+1-+1-24 +

2n-1 2n is finite for even values, but infinite for odd values of the number of terms taken.

+

+

444. A series, all of whose terms have the same sign, must be either convergent or divergent.

For if, for some forms of f and d, the sum of f (r) terms differs from that of $ (r) terms by a quantity whose limit-when p is oo -is not zero, the series must be infinite for all values of n, when n is infinite. Compare Art. 449.

But if, for all forms of f and $, the sum of f (r) terms differs from that of $ (r) terms by a quantity whose limit—when r is op -is zero, the series cannot have more than one finite limit.

445. If a series, all of whose terms have the same sign, is convergent, the series obtained from it by changing the sign of any of the terms will be convergent.

For the new series cannot have a larger arithmetical value than the old. And if, in the new series the difference between f(r) terms and $ (r) terms—when r is oo — were not zero, it could not be zero in the old : hence, as shown in the last article, the old series would be divergent, which is contrary to the hypothesis.

Similarly, if a series, some of whose terms are different in sign, is divergent, the series obtained by making the sign of every term the same will be divergent.

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Theorems on Alternately signed series. 446. (I.) A series which is arranged in groups alternately signed will be finite for all values of n if every group is arithmetically equal to or less than the preceding. Let

U1 - U2 + Uz U4+... be the given series, where un is always positive and = or > Un +1.

The sum

= (Uy Un) + (Uz wa) + and is therefore positive: and = U1 -(Uz - Uz)-(un - Ug) - ... and is therefore algebraically less than 27

Hence the series must be finite for all values of n.

If, however, the limit of the nth group is not zero, the sum of n groups will always differ from that of n 1

groups,

when n is infinite.

But we see by the first of the above forms of bracketing, that the series is equivalent to a series all of whose terms are positive. Hence, by Art. 444, if the limit of the nth term is zero, there can be only one limit of the sum; hence the series is convergent.

447. (II.) Hence, a series which is arranged in groups alternately signed will be finite for all values of n, if every group is arithmetically greater than the preceding but less than some finite quantity. For let every group be less than the finite quantity a.

Then - U+ U2 - Uz + (a – v.) +(a V2) – (a – vz) +

= V1 - V2 + V3 — where V1, V2, V3... are continually decreasing. Hence, by the last theorem, the series is finite for all values of n.

(III.) A series, which is arranged in groups alternately signed, is divergent, if the ratio of every group to the preceding is greater than some quantity greater than 1.

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Then Unt1 > knu,; which can be made greater than any assignable quantity by increasing n.

.. even if the sum of n terms were finite, the sum of n + 1 terms would be infinite.

Also – Un-1 + Un > (k-1) Un-1 =0 when n is o.

Hence, into whatever pairs we group the series, it is always infinite.

Thus, the series is divergent.

Theorems on Series of same sign throughout.

449. (I.) A series, all of whose terms have the same sign, is divergent, if every term is equal to or greater than the preceding.

For in this case the sum of n terms of

+ Un

U1 + 12 + is equal to or greater than n. Uz; and is, therefore, infinite when n is infinite.

The typical series here is that in which the general or nth term is aciwhere x is = or > 1.

450. (II.) A series, all of whose terms have the same sign, is convergent, if the ratio of every term to the preceding is less than some quantity less than unity.

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(1 – k) (U1 + Uz + Uz + ...) = Uy kui + U, kuz + a finite series, by Art. 446, since Unt1 <kun, and kun < Un.

Hence Uy + U2 + Uz + ... is finite, and .. convergent.

The typical series here is that in which the general or nth term is x", where x is < 1.

The preceding two theorems say nothing of the case in which the ratio of each term to the preceding is less than unity, but approaches unity by a quantity whose limit is zero.

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451. (III.) A series, all of whose terms have the same sign, is divergent, if every term multiplied by its 'serial number' is equal to or greater than the preceding term multiplied by its serial number.'

Let (n + 1) Un+1 = nun Then, observing that every term is less than the preceding but approaches indefinitely near to it, we see that

Uz + U4 > 204, i.e. > U2,

Uz + Up + xy + Ug > 4ug, i.e. > Un, and so on.

Hence, by (I), the series is infinite.
A fortiori, if (n+1) Unt1 > nun, the series is divergent.
These conditions may be expressed as follows:

Un+1
31 or n

-1) = 1. n (un Un+1) The typical series here is that, in which the general or nth term is nu where x = or > -1.

1
Thus the series 1 + 1 + } + 4 + + ... is divergent.

Un
Unt1

+

+
n

452. (IV.) A series, all of whose terms have the same sign, is convergent, if throughout

Un+1

<

+ Un

<k, where k < 1.
n (Un – Un+1)
For here

(1 – k) (Uz + Uz + Uz + + un) = U1 + U2 +

Uz
-k{(U, Un) + 2 (u, - U3) + ... + (n − 1) (Un-1 - Un) + nun}.

(– This is clearly less than 4: for u, <k (U, Un), Uz <2k (U9 U3), and so on.

Hence the series Uy + U2 + Uz + is convergent.
This condition may be written in the form

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