Question

In Exercises \(31-34,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-c)^{n}}{n c^{n}} $$

Short answer

The interval of convergence for the given power series is (c-|c|, c+|c|), excluding the endpoints. For endpoints inclusion/exclusion, separate checks for convergence at \(x = c - |c|\) and \(x = c + |c|\) must be applied.

Step-by-step solution

Apply The Ratio Test

To determine the interval of convergence, apply the Ratio Test, which involves taking the limit of the absolute ratio of the (n+1)th term to the nth term as n approaches infinity. Here, that would become: \\[ \\lim_ {n \to \infty} \left| \frac{(-1)^{n+2}(x-c)^{n+1}}{(n+1) c^{n+1}} \times \frac{n c^{n}}{(-1)^{n+1}(x-c)^{n}} \right| \\] \which simplifies to: \\[ \\lim_ {n \to \infty} \frac{n|x-c|}{(n+1)|c|} \\] \For the series to converge this limit must be less than 1.

Determine Convergence Interval

Solving the inequality \( \frac{n|x-c|}{(n+1)|c|} < 1 \) as n approaches infinity gives |x-c| < |c|. This implies that -|c| < x-c < |c|. Hence, the interval of convergence excluding endpoints would be (c-|c|, c+|c|).

Check The Endpoints

For the endpoints c±|c|, you must substitute them into the power series and test for convergence. If the series converges at the endpoint, it is included in the interval, otherwise it isn't. Substitute \(x = c - |c|\) and \(c + |c|\) and apply appropriate tests (like the Alternating Series Test) to check for convergence.