Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/248



We may then write down the following

General directions for finding the interval of convergence of the power series,

Write down the series formed by coefficients, namely,

Calculate the limit

. Then the power series (A) is

.  When $$L = 0, \pm|\tfrac{1}{L}| = \pm\infty$$ and the power series is absolutely convergent for all values of $$x$$. Find the interval of convergence for the series

Solution. First step. The series formed by the coefficients is

By I the series is absolutely convergent when $$x$$ lies between $$-1$$ and $$+1$$.

By II the series is divergent when $$x$$ is less than $$1$$ or greater than $$+1$$.

By III there is no test when $$x = \pm 1$$.