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

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We may then write down the following

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

(A)

First Step. Write down the series formed by coefficients, namely,

Second Step. Calculate the limit

Third Step.. Then the power series (A) is

I. Absolutely convergent for all values of lying between.
and
II. Divergent for all values of x less than or greater than .
III. No test when ; but then we substitute these two values of

in the power series (A) and apply to them the general directions in §141.

Note. When and the power series is absolutely convergent for all values of .

Illustrative Example 1. Find the interval of convergence for the series

(B)

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

(C)
Second step.
Differentiating,
Differentiating again,
Third Step

By I the series is absolutely convergent when lies between and .

By II the series is divergent when is less than or greater than .

By III there is no test when .