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Whence, dividing by n.v-w, and assuming xy,

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.. Multiplying by v, and putting 1+x for v",

m

n

••v=m=1+x{a+2bx+3cx2+4dx3+, &c.}=

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Whence it appears that the formula is correct whether the index be positive or negative. This is Newton's celebrated binomial theorem.

94. By the aid of this theorem, a binomial may be raised to any power, or evolved to any root by simple substitution.

Several examples of the use of this theorem, when the index is a whole positive number, are given in section

second.

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4. Required the value of (a—b)3 in a series.

2

2b 362

8b3

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X3 X3 95. Given z=x+ + +. + 2 2.3 2.3.4 2.3.4.5

finity, to find x in terms of z.

Assume xaz+bz2+cz3+dz++, &c.]

X5

&c. to in

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24

3

Whence, x=x++, &c. where the law. of

2

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3. Required to revert the general series x=ay+by+

cy3+dy+, &c.

Assume y=Ax+Bx2+Cx2+Dx, &c. Then

ay=Aax+Bax2+ Cax3+Dax1, &c.

bA2x2+2bABx3+(B2b+2ACb)xa,&c.
cA3x3+3AoBcx4, &c.

=0.

by2=

cy3=

dy+==

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SUMMATION OF SERIES.

96. Infinite series are sometimes of such nature, that a quantity can be found, to which the series continually approximates, and to which, without attaining perfect equality, it arrives more nearly than by any assignable difference. The quantity to which the series, by continued extension, thus approximates, is called the sum of the infinite series. Thus we say, .3333, &c. to infinity =; for no number less than can be assigned, which the series may not, by extension, be made to exceed.

Required the sum of 1+x+x+x to infinity, x being supposed 1.

Put y=1+x+x2+x3, to infinity.

:. xy=x+x2+x3, &c. to infinity.

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2.- Required the sum of 1+2x+3x2+4x3+5x1+, &c. to infinity, supposing x1.*

Put y=1+2x+3x2+4x3+5x1, &c.

:.-2xy=2x-4x2-6x3-8x+, &c.

And xy=x2+2x3-3x2, &c.

Adding these three equations together

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* The character > is used to express inequality, the opening being presented to the greater quantity.

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