Hence also in the series a ̧ + a ̧ x + a2x2 + in inf. such a

......

COR. value may always be given to x, that the first term is greater than the sum of all the other terms. This value of x will be any quantity less than

387. If each succeeding term of a decreasing Infinite Series bear an invariable relation to a certain number of the preceding terms, the series is called a Recurring Series, and its sum may be found.

Let a + bx + cx+ &c. be the proposed series; call its terms A, B, C, D, &c. and let

CfxBgx2 A, D = fx C + gx2 B, &c.

=

where f g is called the scale of relation; then, by the supposition,

and, if the whole sum A + B + C + D + &c. in inf.

S = A+B+fx.(S – A) + gx2.S,

or S-fxSgx2 S = A + B -fxA;

In the same manner, if the scale of relation be ƒ + g + &c. to n terms, the sum of the series is

A+B+C...to n terms-fx {A+B...to n−1 terms} -gx2 {A+...ton-2 terms} -&c

1 - fx - gx2 - hx3 to (n + 1) terms.

...

Ex. 1. To find the sum of the infinite series 1 + 3x + 9x2 + &c. when is less than

Ex. 2.

To find the sum of the infinite series 1 + 2x + 3x2

If x = or > 1, the series is infinite; yet we know that it arises from the division of 1 by (1 − x)2, and the sum of ʼn terms may be determined.

The series after the first n terms becomes

+2

(n + 1) x" + (n + 2) x2+1 + (n + 3) x" +2 + &c.

in which the scale of relation, as before, is 21; and therefore the series arises from the fraction

COR. If the sign of a be changed, 1 − 2 x + 3x2 - &c. to n terms

1= (n+1) =n+1

(1 + x)2

where the upper or lower sign is to be used, according as n is an even or odd number.

Ex. 3. To find the sum of n terms of the series 1+3x+5x2+7x3+&c. Suppose f+g to be the scale of relation;

then 3f + g = 5, and 5ƒ + 3g = 7; hence ƒ = 2, and g =

= -1;

and, by trial, it appears that the scale of relation is properly determined;

1

(2n+1)x+(2n + 3) x2+1 + (2n + 5) x2 +2 + &c.

hence 1+ 3x + 5 x2 + 7x3 + &c. to n terms,

Ex. 4. To find the sum of 1 + 2x + 3x2 + 5 x3 + 8x1 + &c. in inf. when the series converges.

In this case the scale of relation is 1+ 1, and consequently the sum is

Ex. 5.

- 2 x + 3 x2 − 5 x3 + &c. in inf

=

1 + x x2

To find the sum of n terms of the series

The scale of relation is 21; therefore the sum in inf. is

.. (n − 1) x + (n − 2) x2 + (n − 3) x3 + &c. to n terms,

=

COR. Hence

(n − 1) x

(n-2) x2

+ &c. to n terms,

Ex. 6. To find the sum of n terms of the series

12+22x + 32x2 + 42x2 + &c.

Let the scale of relation be f+g+h; then

9f+ 4g + h = 16,

16f9g+4h = 25,

25f+16g+9h = 36.

From these equations we obtain ƒ = 3, g= 3, h = 1, which values, when substituted, produce the successive terms of the proposed series; therefore

series in inf. when x is less than 1.

After the first n terms the series becomes

(n + 1)2 x2 + (n + 2)2 x2+1 + (n + 3)2 x2+2 + &c.

of which the sum is

(n + 1)2 x” — (2n2 + 2 n − 1) x2+1 + n2x2+2

(1 − x)3

and consequently the sum of n terms of the series is

On this subject the reader may consult De Moivre's Misc. Analyt. p. 72; and Euler's Analys. Infinit. C. XIII.

LOGARITHMS.

1

388. DEF. If there be a series of magnitudes

the indices,

α-3

a", a', a2, a3, ... a*; a ̄1, a ̄2, a ̃3, ... a ̄",

are called the measures of the ratios of those magnitudes to 1, or the Logarithms of the magnitudes, for the reason assigned in Art. 230. Thus, the Logarithm of any number n, is such a quantity, that a = n.

Here a may be assumed at pleasure, and is called the base; and for every different value so assumed a different system of logarithms I will be formed. In the common Tabular logarithms a is 10, and consequently 0, 1, 2, 3, ... x, are the logarithms of 1, 10, 100, 1000, ... 10 in that system.

389. COR. 1. Since the tabular logarithm of 10 is 1, the logarithm of a number between 1 and 10 is less than 1; and, in the same manner, the logarithm of a number between 10 and 100 is between 1 and 2; of a number between 100 and 1000 is between 2 and 3; &c.

These logarithms are also real quantities, to which approximation, sufficiently accurate for all practical purposes, may be made.

Thus, if a be the logarithm of 5, then 10 = 5; let be substituted for a, and 103 is found to be less than 5, therefore is less than the logarithm of 5; but 10 is greater than 5, or is greater than the logarithm of 5; thus it appears that there is a value of a between and 2, such that 10 = 5; the value set down in the Tables is 0.69897, and 10069897 = 5, nearly.

390. COR. 2. Since ao 1, bo

=

1, &c. in any system the logarithm of 1 is 0. Also since a1 = a, the logarithm of the base is always 1.

The method of finding the logarithms of the natural numbers, or forming a Table*, is explained in Treatises on Trigonometry. See Snowball's Trigonometry, Appendix 1.

DEF. If n be any number, log.n signifies the logarithm of n to base a; and log n the logarithm of n to any base.

* Tables of Logarithms have been lately published in a very cheap and convenient form by Taylor and Walton, London, under the superintendence of the Society for the Diffusion of Useful Knowledge.