lies between the greatest and least of the fractions
and is therefore a finite quantity, L say ;
Hence if one series is finite in value, so is the other; if
one series is infinite in value, so is the other; which proves the proposition.
478. Auxiliary Series. The application of the principle of the preceding article is very important, for by means of it we can compare a given series with an auxiliary series whose convergency or divergency has been already established. he series discussed in the next article will frequently be found useful as an auxiliary series.
479. The infinite series is always divergent except when p is positive and greater than 1.
Case I. Let p> 1.
The first term is 1; the next two terms together are less than 2 2^p ; the following four terms together are less than 4 4^p ; the following eight terms together are less than 8 8^p; and so on. Hence the series is less than that is, less than a geometrical progression whose common ratio 2 2^p is less than 1, since p>1; hence the series is convergent.
Case II. Let p=1.
The series now becomes .
The third and fourth terms together are greater than 2 4 or 1 2; the following four terms together are greater than 4 8 or
