decrease. For example, if x=99 100, then 1 1-x= 100, and the terms do not begin to decrease until after the 100th term.
475. Fourth Test. An infinite series in which all the terms are of the same sign is divergent if from and after some fixed term the ratio of each term to the preceding term is greater than unity, or equal to unity.
Let the fixed term be denoted by u_1. If the ratio is equal to unity, each of the succeeding terms is equal to u_1, and the sum of n terms is equal to nu_1; hence the series is divergent.
If the ratio is greater than unity, each of the terms after the fixed term is greater than u_1, and the sum of n terms is greater than nu_1; hence the series is divergent.
476. In the practical application of these tests, to avoid having to ascertain the particular term after which each term is greater or less than the preceding term, it is convenient to find the limit of u_n u_n-1 when n is indefinitely increased; let this limit be denoted by l.
If l<1, the series is convergent. [Art. 473.]
If l>1, the series is divergent. [Art. 475.]
If l=1, the series may be either convergent or divergent, and a further test will be required; for it may happen that u_n u_n-1<1, but continually approaching to 1 as its limit is indefinitely increased. In this case we cannot name any finite quantity r which is itself less than 1 and yet greater than l. Hence the test of Art. 473 fails. If, however, u_n u_n-1>1, but continually approaching to 1 as its limit, the series is divergent by Art. 475.
We shall use “Lim u_n u_n-1” as an abbreviation of the words “the limit of u_n u_n-1" when n is infinite.”
