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372 ALGEBRA. 461. Value of Any Term. In the series

Cy + ye + ae? + 4593 4+ ++,

by taking x small enough we may make any term as large as we please compared with the sum of all that follow it; and by taking x large enough we may make any term as large as we please compared with the sum of all that precede it.

(i.) The ratio of any term, as anxn, to the sum of all that follow it is

= ye” on Cy Cypg rt? A gy Bt? nee Any ® + yy + oe

When x is indefinitely small, the denominator can be made as small as we please; that is, the fraction can be made as large as we please.

(ii.) Again, the ratio of the term anxn, to the sum of all that precede it is ;

= = = > or 5 Cy a Fy, 8? bn af ya? Ho where y = 1 x

When x is indefinitely large, y is indefinitely small; hence, as in the previous case, the fraction can be made as large as we please.

462. The following particular form of the foregoing proposition is very useful.

In the expression

Cy” yO" ore HE +E Cty,

consisting of a finite number of terms in descending powers of x, by taking x small enough the last term a_0 can be made as large as we please compared with the sum of all the terms that precede it, and by taking a large enough the first term anxn can be made as large as we please compared with the sum of all that follow it.

Ex. 1. By taking n large enough we can make the first term of n4 - 5n3 - 7n + 9 as large as we please compared with the sum of all