Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/247



Show that the following ten series are convergent:

Show that the following four series are divergent :

A series of ascending integral powers of a variable, say $$x$$, of the form


 * {| style="width:100%"


 * align="center" |$$a_0 + a_1x +a_2x^2 + a_3x^3 + \cdots$$
 * }
 * }

where the coefficients, $$a_0, a_1, a_2, \cdots$$ are independent of $$x$$, is called a power series in x. Such series are of prime importance in the further study of the Calculus. In special cases a power series in $$x$$ may converge for all values of $$x$$, but in general it will converge for some values of $$x$$ and be divergent for other values of $$x$$. We shall examine (A) only for the case when the coefficients are such that

where $$L$$ is a definite number. In (A)

Referring to tests I, II, III, in §141, we have in this case $$p = Lx$$, and hence the series (A) is