be less than the corresponding terms of the second, for the sum of a finite number of terms of any series must be finite. Ex. Shew that the series 1 + vergent. From the sixth term onwards, each term is less than the corre 45 46 55 525 sponding term of the series + + .... Hence the series beginning at the sixth term is convergent, and therefore the whole series is convergent. 266. Theorem II. If the ratio of the corresponding terms of two series be always finite, the series will both be convergent or both divergent. Ир between the greatest and least of the fractions [Art. 113]. Hence U: V is finite. It therefore follows that if U is finite so also is V, and if U is infinite so also is V. 8r.2 8r 1 is equal to which is > 1 and 8 for all values of r. Now we have (r+1) (r+2) 267. Theorem III. A series is convergent if, after any particular term, the ratio of each term to the preceding is always less than some fixed quantity which is itself less than unity. Let the ratio of each term after the 7th to the preceding term be less than k, where k < 1. Hence the sum of the series beginning at the 7th term is finite, and the sum of any finite number of terms is finite; therefore the whole series must be convergent. 268. Theorem IV. A series is divergent if, after any particular term, the ratio of each term to the preceding is either equal to unity or greater than unity. First, let all the terms after the 7th be equal to u,; then u,+1+r+2 +...+ Un+r = Nu,, and nu can be made greater than any finite quantity by sufficiently increasing The series must therefore be divergent. n. Next, let the ratio of each term, after the 7th, to the preceding term be greater than 1. Then u+u, Ur + 2 > Ur + 1 > Ur, &c. Hence ur+1+Ur + 2 + therefore be divergent. 1 ... +un+r> nu; the series must 2 22 23 2 3 4 Ex. 1. In the series + + + + which is greater than 1; the series is therefore Ex. 2. In the series 12+ 22x + 32x2+ 2 +1). Now, if a be less than 1, and any fixed quantity k that is 1+ х be chosen between x and 1, the test ratio will be less than k for all terms after the first which makes Hence the series is convergent if x < 1. If x = 1 the series is 12+22+32 + which is obviously divergent, and if x>1 the series is greater than 12+22+32 + ..... Thus the series 12+22x+32x2+..... is divergent except when x is less than unity. 269. When a series is such that after a finite number Un of terms the ratio U+1 is always less than unity but Ип becomes indefinitely nearly equal to unity as n is indefinitely increased, the test contained in Theorem III. fails to give any result; and in this case, which is a very common one, it is often difficult to determine whether a series is convergent or divergent. Hence, if k be positive, the test ratio is less than unity, but becomes more and more nearly equal to unity as n is increased. We cannot therefore determine from Theorem III. whether the series in question is convergent or divergent. 1 1 1 270. To shew that the series + + + is con 1* 2* + 3 * 3* ... vergent when k is greater than unity, and is divergent when k is equal to unity or less than unity. First, let k be greater than unity. Since each term of the series is less than the preceding term, we have the following relations: 1 1 2 + < 1 whose common ratio, is less than unity, since k > 1. 2*-1, Hence the given series is convergent. Next, let k = 1; then we can group the series as follows: therefore, as each group of terms in brackets is greater than, the given series taken to 2" terms is greater than 1+1+1+1+...... taken to n+1 terms, that is, greater than 1+n, which increases indefinitely with n. 1 1 1 Hence 1 +2+3+...... is divergent. Lastly, let k be less than unity; then each term of the 1 1 series + + is greater than the corresponding 1* 2 ...... 1 1 term of the divergent series + + ; the series is therefore divergent when k < 1. 271. The convergency or divergency of many series can be determined by means of Theorems I. and II., using the series of the last Article as a standard series. The method will be seen from the following examples. Ex. 1. Is the series whose general term is divergent? Ex. 2. Is the series whose general term is 272. We have hitherto supposed that the terms of the series whose convergency or divergency was to be determined were all of the same sign. When, however, some terms are positive and others negative, we first see whether the series which would be obtained by making all the signs positive is convergent; and, if this is the case, it follows that the given series is also convergent; for a convergent series, all of whose terms are positive, would clearly remain convergent when the signs of some of its terms were changed. If, however, the series obtained by making all the signs positive is a divergent series it does not necessarily follow that the given series is divergent. For example, it will be proved in the next Article that the series-+-+... is convergent, although the series ++++... is divergent. 273. Many series whose terms are alternately positive and negative are at once seen to be convergent by means of |