LIMITS AND SERIES 23 8. Power series. In most of the series studied so far the terms were constant. The most important series, however, are those in which the terms contain some variable, and of these the most important are the so-called power series. A power series is a series of the form "a + V + ^ + Vs +"'> i.e. a series whose terms are positive integral powers of a vari- able, each multiplied by a constant, and in which the terms are arranged according to ascending powers of the variable. The tests obtained for series with constant terms apply to power series also, but it is important to notice that in general the char- acter of the series depends upon the value that is given to the variable. Indeed, the problem in the case of power series usually takes the form, to find those values of the variable for which the series will be convergent. For example, we know that the series l-\-x+s?+x^+ is convergent for all values of x satisfying the condition -l<a;<+l, or, briefly, |a-|<l. We have not ascertained its character when x == 1 or x = — 1, but for all values of x which are less than — 1, or greater than +1, the series is divergent. One great value of power series lies in the fact that they fur- nish an easy means of computing the numerical values corre- sponding to special values of the variable for many functions where such computation would otherwise be extremely difficult or even impossible. For example, the cosine of x may be defined by the series „ , 1 x , x cos.-=l-^+^-3-4----, where the unit of measure for x is the radian which is the equiv- alent of 57°.2957795+. Suppose it is required to find the cosine of one radian. By definition,