196. The powers of a fraction less than unity continually decrease; thus, the square of ⅖, or ⅖ × ⅖, is less than ⅖, being only two-fifths of it. This decrease continues without limit; that is, there is no quantity so small but that some power of ⅖ is less. For if
| 5 | = | x, | |
| 2 | |||
| 2 | = | 1 | , |
| 5 | x |
| and the powers of ⅖ are | 1 | , | 1 | , |
| xx | xxx |
and so on. Since x is greater than 1 (195), some power of x may be found which shall be greater than a given quantity. Let this be called m; then 1/m is the corresponding power of ⅖; and a fraction whose denominator can be made as great as we please, can itself be made as small as we please (112).
197. We have, then, in the series
1 r rr rrr rrrr &c.
I. A series of increasing terms, if r be greater than 1. II. Of terms having the same value, if r be equal to 1. III. A series of decreasing terms, if r be less than 1. In the first two cases, the sum
1 + r + rr + rrr + &c.
may evidently be made as great as we please, by sufficiently increasing the number of terms. But in the third this may or may not be the case; for though something is added at each step, yet, as that augmentation diminishes at every step, we may not certainly say that we can, by any number of such augmentations, make the result as great as we please. To shew the contrary in a simple instance, consider the series,
1 + ½ + ¼ + ⅛ + ¹/₁₆ + &c.