5 2 1 1 1 --- = _x_, --- = ---, and the powers of ⅖ are ----, -----, 2 5 _x_ _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. Carry this series to what extent we may, it will always be necessary to add the last term in order to make as much as 2. Thus, (1 + ½ + ¼) + ¼ = 1 + ½ + ½ = 1 + 1 = 2 (1 + ½ + ¼ + ⅛) + ⅛ = 2. (1 + ½ + ¼ + ⅛ + ¹/₁₆) + ¹/₁₆ = 2, &c. But in the series, every term is only the half of the preceding; consequently no number of terms, however great, can be made as great as 2 by adding one more. The sum, therefore, of 1, ½, ¼, ⅛ &c. continually approaches to 2, diminishing its distance from 2 at every step, but never reaching it. Hence, 2 is celled the _limit_ of 1 + ½ + ¼ + &c. We are not, therefore, to conclude that _every_ series of decreasing terms has a limit. The contrary may be shewn in the very simple series, 1 + ½ + ⅓ + ¼ + &c. which may be written thus: 1 + ½ + (⅓ + ¼) + (⅕ + ... up to ⅛) + (⅑ + ... up to ¹/₁₆) + (¹/₁₇ + ... up to ¹/₃₂) + &c.