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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 and by writing the nth term

T'a bears a finite ratio to √x (~)*,

in the form

Гь г (a + n)
Ta T (b + n)

th

parable with

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no

13. We will now examine the series given us by the methods of Chap. V.

By (22) page 100 we have

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whence follows the result found above.

1 1 2 B.

2x2

a ' | 2

1

t", it will be found to be com

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Un+1

Un

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+

B B

=

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2.3.4

acă

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Here numerically

ultimately {see (9) page 109},

and thus the series ultimately diverges faster than any diverging geometrical series however large x may be.

&c.,

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 is.so 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 u 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 Thus the zero I is of

divergent in spite of un becoming ultimately zero.

n

1

gives a divergent series;

n

will be convergent if the series

too low an order, since un

1

high order, since un= represents a convergent series. Now the series on n2

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 many value >1 in the rth 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) + ...

F(2), F(3)
+

log 2 log 3

1

a2 + 1

1

222

n2

+

+

F (4)
log 4

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

A method of testing convergence is given by Kummer (Crelle, XIII.), inferior, 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 v vn un such that vnu=0 ultimately, but · Un+1 >0 when n is co. His further Un+1 paper is in Crelle, xvI. 208.

is of a sufficiently

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.

1

a2 + 4

EXERCISES.

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

1

+
+ &c.
a2 +9

In particular shew that if a 1 the numerical value of

=

π

π

the series will lie between the limits

2. The sum of the series

1

1

11+8 + 21+8+ &c.

(where & is positive) lies between

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2nta

3. Examine the convergency of the following serics

1+ e12 + e ̄(11 + 3 ) + e ̄(1+1+1) + &c.,

log 2

1.

2

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1+2+3+ &c.,

1+2+3+ &c.,

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1" 2n

1+ + + &c.,
3nta

and

1

sin x
3

3*

α

'+ (log 3)" +
3)བ

2

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28

28-1'

+ &c.

...

4. Are the following series convergent?

7

9

2n+1

3 5
x+= x2+
5

2 x

where x is real.

2

10

17

n2 + 1

1 + x cos a + x2 cos 2x + ....................... where x is real or imaginary.

and

+ &c.,

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5. The hypergeometrical series

a (a + 1) b (b + 1)
c (c + 1) d (d + 1)

is convergent if x <1 divergent if x > 1.

If

ab

1 + x +
cd

if

7. In what cases is

x

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

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

23

203

+2+3 2.3+ &c.

1.2

x2+ x x + x x + x
x2+1x+12+1

8. Shew that

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1

1 1 + + Uo u1

is convergent if u2+2' ·2un+1+u, be constant or increase with n.

9. If

B

n

n =

น.

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+ &c.

Y

+ + &c.,

10. A series of numbers P1, P2

x2+ &c.

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

finite?

Pn

A log p1 + B'

... are formed by the formula

+ + &c. is convergent.

shew that the series F (p1) + F (p2) + &c., will be convergent

F (2), F(3) log 2 log 3

[Bonnet, Liouville, VIII. 73.]

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12. Shew that if ƒ (x) be such that

xf'(x)
f(x)

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

[Abel, Crelle, III. 79.]

.........

=

ø (1) + $ (2) + $ (3) + ......... $ (1) +mp (m) + m2p (m2) + ....

vergent or divergent together.

of the original series is

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when x = 0, the series u1+u2+ ................ and ƒ (u') +ƒ (u2) + ... converge and diverge together.

......

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

converge and diverge

1

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

[Paucker, Crelle, XLIII. 138.]

15. If α +α1x+a,x2 + &c. be a series in which a, a, &c., do not contain x and it is convergent for x = 8 shew that it is convergent for x <8 even when all the coefficients are taken with the positive sign.

m being positive are con

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

1

the case of u2= $(n) cos n0. Ex. $ (n) =

when the sum

n

log (2-2 cos x).

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