it therefore follows from Art. 270 that the given series is convergent if k>1 and divergent if k1.

336. We shall conclude with the two following tests of convergency which are sometimes of use, referring the student to Boole's Finite Differences and Bertrand's Differential Calculus for further information on the subject.

337. Theorem. A series is convergent when, from and after any particular term, the ratio of each term to the preceding is less than the corresponding ratio in a known convergent series whose terms are all positive.

be

For let the series, beginning at the term in question,

U = U1 + U2 + Uz +... Un

and the known convergent series, beginning at the same term, be

Then, since <

vz 2

Vr+1 for all values of r, we have

Ur+1
Ur Vr

U z + V 1
Из Иг
U2 U1

1

Из Из Иг
+01
Из Из И1

+

Hence as V is convergent, U must also be convergent. The given series is therefore convergent, for the sum of the finite number of terms preceding the first term of U must be finite.

We can prove similarly that if, from and after any particular term, u,+1:u, > v and all the terms of Σu, have the same sign; then Eu, will be divergent if Σv, be

r+1

: Vr,

divergent.

338. Theorem. A series, all of whose terms are positive, is convergent or divergent according as the limit of

n+1

n (1 - Uat) is greater or less than unity.

Ип

First suppose a > 1, and let ẞ be chosen between

a and 1.

Then since the limit of n

the limit of n (1 Un+1),

w

there must be some finite value

of n from and after which the former is constantly greater than the latter.

But when a (1") > n (1),

n

Hence, by the previous theorem, Eu, will be convergent if Ev, be convergent; but Σv, is convergent since B>1.

n

Similarly, if a be < 1, and ẞ be taken between a and 1, we can prove that Zu, is divergent if Σv, is divergent, and the latter series is known to be divergent when <1.

n

Un+1 a + n
Ix, the limit of which is x.
Ип
b+n

Hence, either from

the beginning or after a finite number of terms,

Hence the series is divergent if x > 1, and convergent if x <1.

1.

series is convergent when ba> 1 and When b=a+1, the series becomes

a

a

a

+ ......9

+
b b+1 b+2

[These are the results arrived at in Ex. 2, Art. 333.]

EXAMPLES XXXIV.

Find the sum of each of the following series to n terms, and when possible to infinity :

:

2.

+

14 14.16 14.16.18

+

Find, by the method of differences, the nth term and the sum of n terms of the following series :—

(vi) 1+2+29 + 130 +377 + 866 + 1717 +

3.

Find the generating function of each of the following series on the supposition that it is a determinate recurring series :

(v) 12 + 2x + 32x2 + 42x3 + 52x2 +• 631⁄23 +

4. Find the nth term, and the sum of n terms of the following recurring series:

5.

Find the nth term of the series 1, 3, 4, 7, &c.; where, after the second, each term is formed by adding the two preceding terms.

6. Determine a, b, c, d so that the coefficient of x" in a + bx + cx2 + dx3

the expansion of

(1 − x)1

may be (n + 1)3.

7. Shew that the series 1′+2*x + 3′x2 + 4x3 +

0

expansion of an expression of the form a +a,x+... ax

and that a-s 0;

(1 − x)TM+ 1

8. Find the sum to infinity of the recurring series

2+5x+9x2 + 15×3 + 25x2 + 43x5 + ...

supposed convergent, it being given that the scale of relation is of the form 1 + px + qx2 + rx3. Shew that the (n + 1)th term of the series is (2′′ + 2n + 1) x”.

9. Find the sum to infinity of the series

1+ 4x + 11x2 + 26x3 + 57x2 + 120x3 +

+

(1 + n) (1 + 2n)

p+ n

(p + n) (p + 2n) 1 (p + n) (p + 2n) (p + 3n)

m+ 1

+

m

...

+ to infinity =

m

1

(n − 1) (n − 2)

1

(m + 1) (m + 2) * (m + 1) (m + 2) (m + 3)

if m + n be positive, or if n be a positive integer.

+