Problem 32 - Essential Calculus

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.) $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-c)^{n}}{n c^{n}} $$

Step-by-Step Solution

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 1: 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.

Step 2: 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|).

Step 3: 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.

Key Concepts

Power Series

A power series is an infinite sum of terms in the form \(a_n(x-c)^n\) where each term involves a power of the form \((x-c)^n\). This general form features a "center" \(c\), around which the function represented by the series converges. Power series are significant because they can approximate functions that are otherwise difficult to express or compute.

• The series has coefficients \(a_n\), and \(x\) is the variable.
• They are used in many areas of calculus, including solving differential equations and representing analytic functions.

When we talk about the convergence of a power series, we are interested in the range of \(x\) values for which the series converges to a finite sum. This range is known as the interval of convergence, and finding it involves tests such as the Ratio Test.

Ratio Test

The Ratio Test is a tool used for determining the convergence of an infinite series like a power series. The basic idea is to look at the ratio of the (n+1)th term of the series to the nth term. If this ratio is consistently less than one as \(n\) tends to infinity, the series converges.
Here's how it typically works:

• Compute the absolute value of the ratio of successive terms: \(\lim_ {n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
• If this limit is less than 1, the series converges absolutely.
• If the limit is greater than 1 or the limit does not exist, the series diverges.
• If the limit equals 1, the test is inconclusive.

In our specific exercise, this translates into determining when \(\frac{n|x-c|}{(n+1)|c|} < 1\). This inequality helps find the interval, \(|x-c| < |c|\), within which the series converges.

Convergence at Endpoints

Once we find an interval of convergence from the Ratio Test, the next step is to check whether the series converges at each endpoint. Often, the test used initially might not give results for these specific values at the boundaries.
Endpoint checking involves:

• Substituting the endpoint values back into the series.
• Evaluating the convergence of the series at these specific values using different tests (if necessary).
• Comparing the series at \(x = c - |c|\) and \(x = c + |c|\) to determine if these points should be included in the interval.

For example, if you substitute \(x = c - |c|\) and apply an appropriate test, such as the Alternating Series Test, it helps decide if the endpoint is part of the convergence interval. If the series converges at these points, they are included; otherwise, they are excluded.

Alternating Series Test

The Alternating Series Test is used to determine the convergence of series whose terms alternate in sign. These are typical whenever you see terms like \((-1)^n\), which changes sign for each consecutive term in the series.
For an alternating series \(\sum (-1)^n b_n\):

• The series converges if the absolute values of the terms \(b_n\) decrease: \(b_{n+1} \leq b_n\) for all \(n\) beyond a certain point.
• The limit of \(b_n\) as \(n\) approaches infinity should be zero.

The Alternating Series Test is especially useful when testing convergence at the endpoints of a power series. If the series converges at these endpoints based on the alternating nature and decreasing sequence of terms, then it bounds the interval of convergence.