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It is by no means necessary to resort to the criteria of system (C) in this case. From (13) page 94 we learn that

Гь г (a+n)
in the form

t", it will be found to be com-
Tar (b + n)
parable with whence follows the result found above.

0-a n

13. We will now examine the series given us by the methods of Chap. V. By (22) page 100 we have 1

1 1 2 B, 2.3.4 B C

2.0" x 2 + 200 14
1 1





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na Here numerically

xr7 ultimately {see (9) page 109}, and thus the series ultimately diverges faster than any diverging geometrical series however large x may be.

As it stands then our results are utterly worthless since we have obtained divergent series as arithmetical equivalents of finite quantities and in order to enable us to approximate to the numerical values of the latter. We shall therefore recommence the investigations of Chap. V, finding expressions for the remainder after any term of the expansion obtained, so that there will always be arithmetical equality between the two sides of the identity, and we shall be able to learn the degree of approximation obtained by examining the magnitude of the remainder or complementary term.

14. The solution of the problem of the convergency or divergency of series that has been given complete that it is scarcely possible to imagine how a case of failure could arise. But we have not only obtained a test for con.



vergence, we have also classified it. Let us consider for a moment any infinite series. Its nth term Un must vanish, if the series is convergent, but it must not become a zero of too low an order; otherwise the series will be

1 divergent in spite of Un becoming ultimately, zero. Thus the zero is of 1

1 too low an order, since Un = gives a divergent series; is of a sufficiently

na 1 high order, since

Un represents a convergent series. Now the series on

n? page 128 give us a classified list of forms of zero. The zeros of any one form are separated by the value m=1 into those that are of too low an order for convergency and those that are not. But between any zero value that gives convergency and that corresponding to m=1 (which gives divergency) come all the subsequent forms of zero. Series comparable with the series produced by giving m any value >1 in the oth class converge infinitely more slowly than those with a greater value of m, but infinitely faster than any similarly related to the (r+1)th or subsequent classes, whatever value be given to m in the second case. Thus we may refer the convergency of any series to a definite standard by naming the class and the value of m of a series with which it is ultimately, comparable.

15. Tchebechef in a remarkable paper (Liouville, xvII. 366) has shewn that if we take the prime numbers 2, 3, 5... only, the series

F(2)+F(3)+F(5) + ... will be convergent if the series

F(2), F (3)

F (4)

+ log 2log 3

log 4

+... is convergent. Compare Ex. 10 at the end of the Chapter.

A method of testing convergence is given by Kummer (Crelle, XIII.), in. ferior, of course, to those of Bertrand, &c., but worthy of notice, as it is closely analogous to his method of approximating to the value of very slowly converging series (Bertrand, Diff. Cal. 261). It is by finding a function on such that v„Un=0 ultimately, but on Un

> when n is oo. His further paper is in Crelle, XVI. 208.

We shall not touch the question of the meaning of divergent series ; De Morgan has considered it in his Differential Calculus, or an article by Prehn (Crelle, XLI. 1) may be referred to.



EXERCISES. 1. Find by an application of the fundamental proposition two limits of the value of the series

1 1 1

ü +1 a* + 4 a* +9


+ &c.

In particular shew that if a=1 the numerical value of the series will lie between the limits and

2 4

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3. Examine the convergency of the following serics

1 +61 +e (1+1) + e*(1+3+) + &c.,
1+et+2 +1+3) + €-(*3*3) + &c.,

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4. Are the following series convergent ? 7 9

2n +1 x + = x +


where x is real. 10 17

n+1 1 + x cos a + x* cos 2% + ........... where x is real or imaginary.

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5. The hypergeometrical series


a (a + 1) 6 (6+1) cd

+ &c. c(c + 1) d (d + 1) is convergent if w<1 divergent if x> 1.

If xf = 1 it is convergent only when c+d-a-b> 1.


6. For what values of ac is the following series convergent ?

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is convergent if Untz — 24942 tu, be constant or increase with n.

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shew that the series converges only when a<1, or when a=1, and ß>1.

10. A series of numbers Pa, Pg ... are formed by the formula

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shew that the series F (p) + F(P) + &c., will be convergent

F (2), F (3) if

+ &c. is convergent. log 2' log 3

[Bonnet, Liouville, VIII. 73.1

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11. Shew that the series

Qo + ax + a, + &c., and an +


+ converge and diverge ao

a,+ a at a, t a. together.

Hence shew that there can be no test-function $ (n) such that a series converges or diverges according as $ (n) + Un does not or does vanish when n is infinite.

[Abel, Crelle, III. 79.] 12. Shew that if f (2) be such that

æf' (a)

f (x) when x=0, the series u, tu, + ...... and f (2) + f (u) + ...... converge and diverge together.

13. Prove from the fundamental proposition Art. 6 that the two series

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$ (1) + (2) +$ (3) + ........

in being positive are con$ (1) + (m) + m'ø (mo) + vergent or divergent together.

14. Deduce Bertrand's criteria for convergence from the theorem in the last example.

[Paucker, Crelle, XLIII. 138.] 15. If a, + a2 + ax + &c. be a series in which a, a, &c., do not contain x and it is convergent for x = - O shew that it is convergent for x < d even when all the coefficients are taken with the positive sign.

16. The differential coefficient of a convergent series remains finite within the limits of its convergency. Examine

1 the case of Un=$(n) cos no. Ex. $ (n) =

when the sum

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of the original series is – log (2 – 2 cos a).

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