The given series may be written in each of the following forms :
(u — te) + (tg — UH) —%)+ ©. . . . A), Wy — (Uy — Us) — (Uy — Us) —(Ugp — UG J— - - - (2).
From (1) we see that the sum of any number of terms is a positive quantity ; and from (2) that the sum of any number of terms is less than u_1; hence the series is convergent.
For example, in the series
1 1 al 1 1
the terms are alternately positive and negative, and each term is numerically less than the preceding one; hence the series is convergent.
471. Second Test. An infinite series in which all the terms are of the same sign is divergent if each term is greater than some finite quantity, however small.
For if each term is greater than some finite quantity a, the sum of the first n terms is greater than na; and this, by taking x sufficiently great, can be made to exceed any finite quantity.
472. Before proceeding to investigate further tests of convergency and divergency, we shall lay down two important principles, which may almost be regarded as axioms.
I. If a series is convergent it will remain convergent, and if divergent it will remain divergent, when we add or remove any finite number of its terms; for the sum of these terms is a finite quantity.
II. If a series in which all the terms are positive is convergent, then the series is convergent when some or all of the terms are negative; for the sum is clearly greatest when all the terms have the same sign.
We shall suppose that all the terms are positive unless the contrary is stated.
