4. It is required to find the square of the series 1 + x + x2 + x3 + x1&c.

Ans. 1+2x+3x+4x+5x+&c.

5. It is required to find the square root of the series 1 + x + x2 + x3 +x*&c.

6. It is required to find the square of the series

Ans. y-2y+3yo — 4y3 + 5y1o&c.

7. It is required to find the 5th power of the

series 2x+3x-4x2 + 5x* _ 6x &c.

8. It is required to find the square root of

3) REVERSION OF SERIES, is the finding the value of the root, or unknown quantity, contained in an infinite series, by means of another infinite series, in which some other quantity only is contained; the method of doing which is as follows;

RULE I.

If the series consist of all the powers of the unknown quantity, as ax + bx2 + cx3 + dx* + ex3&c. =y, substitute the particular values of the coefficients, in the given example, for a, b, c, d, &c. in

the following formula, and the result will be the value of x required.

y ya

x = 2 − b22 + (2b3 — ac)23 — (5b3 — 5abc + a°d)1⁄2 1⁄2 +

a

(14b* — 21ab°c + 3a°c2 + 6aobd — a’e)1⁄2o — (42b3 —

84ab°c + 28a©bc2 + 28a©b°d — 7aˆed — 7a'bé + a1ƒ)2o &c. (n)

Where it is to be observed, that, if y be a large number, the series will often diverge, and consequently, in that case, be of no practical use.

EXAMPLES.

1. Given x + + +

2 3 4

value of x in terms of z.

+ &c., to find the

(n) The formula for the conversion of series, here given, was one of Newton's first improvements in analysis, which he transmitted, in 1676, in a letter to Oldenburgh, at that time Secretary to the Royal Society, with directions to have it communicated to Leibnitz; which letter, together with another on the same subject, was afterwards printed in the Commerium Epistolicum of Collins, 1722.

Arbogast, in his work before quoted, p. 240, has given the reverted series in a form which renders the law of its continuation sufficiently obvious; but as the symbols which he employs are made to denote the operations that are to be performed in deriving the successive, coefficients of the several terms from each other, instead of the coefficients themselves, his theorem cannot be exhibited in this place, for the reasons already mentioned.

It may be farther observed, that the reverted series has been computed and verified in numbers, as it appears, for the first time, by Rubbiani, an Italian analyst, as far as to

or to the

9th term inclusively. See Trigonométrie de Cagnoli, p. 46, 2d edit. where the series is given to the extent abovementioned.

VOL İ.

1

Here a=1, b=1, c=1, d=}, e=}&c. and y=x.

+ (14b* — 21'ab°c + 3a°c2 +6abd — a’e) =

120

-(42b3 — 84ab3c+28abc2 + 28a b'd-7acd-7abe+

2. Given x-x2 + x3 — x2 + x3 − x° +x1&c.=%, to

find the value of x in terms of %.

Ans. x=x+x2 + 83 + ≈1 + ≈3 +≈°&c.

2

4

ყვ y4
+ +

y5 36
+

&c.

8 16 32

y2 2y3 2y1 2ys 32yo 24447&c.=%,

5. Given y ++ + + + +

r 3r2 3r3 314 45r5

to find the value of y in terms of %.

31576

312 3r3 34

+ &c. 4515 31576

RULE II.

If the series consist of the odd powers of the unknown quantity only, as ax + bx3 + cx3 + dx2+ ex &c.y, substitute the particular values of the coefficients, in any given example, for a, b, c, d, &c. in the following formula, and the result will be the value of x, as required.

b22 + (3bo — ac)22 — (12b3 + ad — 8abc)2,1% +

a4

10

(55b* — 55ab°c+10a3bd+5a°c2 — a°e)2 &c.

Where it is to be observed, that, in this case, as well as in the former, the given and reverted series must be both of the same form, or otherwise they are not convertible into each other.

1

2.3.4.5.6.7.8.9 terms of %.

1

2.3

3

1

+

2.3.4.5

x&c.=%, to find the value of x in

d=

1

--

5040'

362880' Y,

6 2.3

Whence

3

3

40 2.4.5

+ (55b* — 55ab°c + 10a3bd + 5a ̊c2 — a3e) = (

2. Given x+x3 + x3 + x2 + x°&c.=y, to find the

value of x in terms of y.

Ans. x=y-y+ 2y3 — 5y1 + 14y°&c.

to find the value of x in terms of z.

When two series are equal to each other, as ax + bx2 + cx3 + dx* + ex3&c. = ay + by2 + 7y3 + dy* + ay'&c. and it is required to find the value of the unknown quantity in one of them, in terms of the unknown quantity in the other, substitute the particular values of the coefficients in the given example, in the place of the known letters, in the following general formula, and the result will be the root, or value required.

(0) It is something remarkable, as has been observed by Maseres, Script. Logarithmici, Vol. III, p. 479, that the complicated series, in Example 4, which is the log. tan. (45° + -*), should have its several terms the same, except with regard to the signs, as those of the reverted series; this being the only instance of such a series yet known.