Example 5: Interval of Convergence

Find the interval of convergence of

We have

By the Ratio Test the series converges for |x + 5| < 4 and diverges for |x + 5| > 4. The radius of convergence is r = 4, and the interval of convergence is centered at -5. We note that

Therefore at |x + 5| = 4,

Thus at x + 5 = 4 and x + 5 = -4 the terms do not approach zero and the series diverges. The interval of convergence is (-9, -1).