LIMITS AND SERIES 19 In general, a finite number of terms may be stricken out from the beginning of a series with constant terms, and the test ap- plied to the resulting series, without any loss of generality, for the sum of the terms stricken out is finite and determinate. Consequently, if the sum of the first n terms remaining after terms are stricken out from the beginning has a limit as n in- creases, the sum of the first n terms of the original series will have a limit as n increases. This last limit will be the sum of the terms stricken out plus the sum of the convergent series that remains after the terms are stricken out. For the purposes of the direct-comparison test it is desirable to have a number of series known to be convergent or divergent to begin with. All geometrical series whose ratio is less than 1 are convergent, while all geometrical series whose ratio is greater than 1 are divergent. The series 1+^+--- was shown in § 5 to be divergent. The series ^ + ^ + ojc + " ' is convergent when k > 1 and divergent when k s 1. The proof is given in most textbooks on algebra. 7. The test-ratio test. The test-ratio test, which is one of the most useful known tests, holds for series having both positive and negative terms. It makes use of the ratio of the general term to the preceding term, viz. ^('-t-l, the so-called test ratio. The test- ratio test is given in one of "its most useful forms by the following THEOREM. Let u^ + u^ + Mg + be an infinite series, and let lim -"±1 =t;
\[m-""-=t; n=» U,
then, if \ t \ * < 1, the series converges; if \ t \ > 1,the series diverges; if\t\=l, the character of the series is undetermined. * As in § 4, the notation 11 \ is used to denote the numerical value of t. 20 MATHEMATICAL THEORY OF INVESTMENT Proof for series having only positive terms. By the conditions of the theorem lim"-^^, «=» «„ and for the case under consideration ( is a positive number.
Suppose first that
t<l.
Itis then possible to find a fixed number r lying between ( and 1, i.e. a number satisfying the condition t<r<l. A set of constant numbers A-i, ^, Zy can then be found such that u^=k^r, u^=k,r\ ^=^r8, , andthe series itself may be written ^r+^r2+y+. Itfollows that after a definite term for which n = m — 1, ""m^ "'111+1 >"'i>i+2 ' ' ' »/ 1e r" +' For, -^=—i:1 , ^n ^"B^ '?/ and, moreover, because the variable —"±1 approaches the limit <, u un the value of "+1 will be less than r, so that "m ^iT^-^ir^ ^r» ~ A ' '
and, from the inequality, ^u^ ^ r k f Now, since the fraction -a±l is less than the proper fraction -i ^ ' ' r the numerator A^+i must be less than the denominator k^, as was asserted. LIMITS AND SEEIES 21 Furthermore, since ^ > ^, +1 > ^m +2 > " '» the terms of the series M»+M»+l+^+2+ , or, what is the same thing, the terms of the series ^rm+Z„^rm+l+^^+2+, are, after the first term, less than the corresponding terms of the geometrical series ^+^m+l+v+2+, which is convergent, since the ratio r is numerically smaller than unity. Consequently, the series U^+tl^+U^+ ' ' '' from which the convergent series «m+M»+l+Mm^.2+ was obtained by striking out the first m — 1 terms, is convergent. In a similar manner, when t> 1, it may be shown that after a certain term M^_i of the series u^+u^+Uy+ the terms %„,., u^,,^, M^+a, will be equal to or greater than the terms of the divergent geometrical series ^ r-'+jfc ,rm'+l+ . Under such conditions the series M^+M^+i+M^+2+ ; and consequently the series Mi+^+Mg+.--, from which it was obtained by striking out m'—l terms, will be divergent. Finally, if t = 1, no conclusion can be drawn, for the limit of the test ratio of the series l+-^+i+---. 22 MATHEMATICAL THEORY OF INVESTMENT which was shown to be divergent, is unity, while, on the other hand, the limit of the test ratio of the convergent series A,1,1,... I2 ' 22 32 is also unity. The theorem is therefore proved for series having only positive terms. The proof-for series having both positive and negative terms is not difficult but cannot well be given here. ILLUSTRATIVE EXAMPLE. It is required to determine the character of the series ^r^rTih^-- The general term is ———————, and the ratio -2-±1 is therefore 1 2 3 n M,
1 2 3 n (n + 1)1
1n+1
Consequently,
1.2.3...n
lim"±l=<=l^n—!———0. »=» "» »=»»+1
Since t = 0, the series is convergent.
EXAMPLES Determine the character of the following series: ' l-S^"^ 1.2.3.4.54" '"' + 12. 22 + 22 32 + 32. 42' 3 3. 3-5 3-5-7 357 (2n+ 1) ' 4 4 7 4 7 10 ' " ' ' 4 7 10 (3 n +1) 4" iTa!+ ir"! + er^ + '"' s,l-2+^^+- ^H^-1--- 6.1+1+1^+.... 8.1+l+l+.... 2468 V2 V3