16 MATHEMATICAL THEORY OF INVESTMENT (3) The sum »„ will approach no limit, either finite or infinite. Such a series is i_ -iii_-|ii_ where the value of «„ is always 1 or 0. If the sum «„ approaches a finite limit S, as in the first case, the series is said to be convergent with sum S, and we write - S=u^+u^+u^+ . In all other cases the series is said to be divergent. A divergent series has no value whatever for the purposes of ordinary com- putation ; whence it follows that the prime question for consid- eration is the determination of the character of the series as to convergence or divergence. A necessary condition for the convergence of an infinite series is that the successive terms themselves should diminish toward zero, for if we denote the sum of all the terms after the rath by R,, we shall have <_ . p " — "n I -"'n- Since lim ^ = S, n= no it follows that lim £^ = 0. n=«> From this it follows that the sum of any number of terms fol- lowing the nth approaches zero as n .increases, and in particular the limit toward which the value of a single term a, approaches as n increases is zero. A single example, viz. the series 1+^44+..., which was shown to be divergent, shows that the condition is not sufficient to insure convergence. No general practical test for the determination of convergence or divergence is known, but there are two, the direct-comparison test and the test-ratio test, which suffice for the determination of the character of a large number of important series. In what follows only series with real terms will be considered. 6. The direct-comparison test. In the direct-comparison test the terms of the series to be tested are compared with the corre- sponding terms of a series whose character is known. This test, LIMITS AND SERIES 17 when used to determine the character of series having only posi- tive terms, may be stated in the form of the following THEOREM. Let u^+ u^+ ziy+ + u^+ be a series of positive terms which is to be tested for convergence or divergence, and let ^+ <^+ a, ++<»„+ be a test series of positive terms whose character is known. Ifthe test series is convergent, and if the terms of the series to be tested are less than, or at most equal to, the corresponding terms of the test series, the series to be tested is convergent, and its sum is not greater than the sum of the test series; if, on the other hand, the test series is divergent, and if the terms of the series to be tested are equal to or greater than the corresponding terms of the test series, the series to be tested is divergent. Proof when the test series is convergent. Let ffll+fli2+"+a»+'" be a series of positive terms which is known to be convergent with sum A, and let A, be the sum of the first n terms. Also let Mi+^+ +"»+ be a series of positive terms to be tested, and let «„ be the sum of the first n terms. "We have to prove that if we have always Un S <»„, 8, approaches a limit which is equal to or less than A. Since every term u is less than, or at most equal to, the cor- responding term a, the sum ^ = M! + "3 ++"» will be less than, or at most equal to, the sum A = a! + ^ ++»„ Butby hypothesis lim A,, = A n= ca and, since the terms of the test series are all positive, ^<A for every value of n. Consequently, the variable «„ is always less than A. Moreover, because the series to be tested contains nothing but positive terms, »„ always increases as n increases. 18 MATHEMATICAL THEORY OFINVESTMENT It follows that s, is a variable which always increases but never takes values greater than a given fixed number A. By Theorem I (§ 4) »» approaches a limit which is equal to or less than A, and the series to be tested is convergent. Proof when the test series is divergent. In this case «„ g A,, and A^ increases without limit. Obviously, »„, which is not less than A' w1^ ^so increase without limit. Consequently, the series to be tested is divergent. ILLUSTRATIVE EXAMPLE. It is required to determine the character of the series ^ ^ __^ ^ 1 + r~2 + n2~3 + T~2~S~i + ''' + T~2~S~n + '' ' For a comparison series, the geometrical series l+J+|5+|s+---+|;+---, whose ratio is ^, and which is therefore convergent, may be taken. Each of the first two terms of the series to be tested is equal to the correspond- ing term of the test series, as may be seen by inspection. Moreover, every term of the series to be tested, after the second, is less than the correspond- ing term of the test series, since T2~3~i~n < ^ for all values of n greater than 2. The series to be tested is therefore convergent. The direct-comparison test is frequently useful in cases where the conditions of the theorem do not hold for the first few terms of the series to be tested and the test series. For example, comparing the series 1+T+^-2+1.23+12.34+"' with the convergent series +2+22+28+24+"'' it is seen that the condition required by the test holds for the first term but does not hold for the next three. If, however, we strike out the first term of the series to be tested, or if we prefix the term 2 to the test series, the difficulty disappears, while the character of both the series in question remains unchanged.