Example 1: Interval Of Convergence

Find the interval of convergence of the power series

This is just the geometric series

1 + bx + (bx)² + ... + (bx)ⁿ + ....

It converges absolutely when |bx| < 1, |x| < 1/b, and diverges when |bx| > 1, x| > 1/b. So the radius of convergence is

r = 1/b.

At x = r and at x = -r the series diverges, because bⁿrⁿ = 1. Thus the interval of convergence is (-1/b, 1/b).

The Ratio Test can often be used to find the radius of convergence of a power series.