Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A power series of the form \(\sum_{k=0}^{\infty} a_{k}(x-2)^{k}\) has a radius of convergence of \(3 .\) (a) For what values of \(x\) can you say with con dence that the series converges? (b) For what values of \(x\) can you say with con dence that the series diverges? (c) For what values of \(x\) are you given inadequate information to determine convergence?

Step-by-Step Solution

Answer

The series converges if \(x\) lies in the interval (-1, 5). The series diverges if \(x < -1\) or \(x > 5\). For the values \(x = -1\) and \(x = 5\), further information is needed to determine if the series converges or diverges.

Step 1: Convergence Interval

Determine the interval of convergence. Since the radius of convergence is 3 and the series is centered at \(x = 2\), the convergence interval is \((2 - 3, 2 + 3)\) which simplifies to \((-1, 5)\). Hence, the series converges for all x in this interval.

Step 2: Divergence Interval

Establish the interval of divergence. If the absolute value of \(x - 2\) is greater than the radius of convergence (3), the series diverges. This gives us \(x < -1\) or \(x > 5\), which are the intervals of divergence.

Step 3: Uncertain Convergence

Identify the values where the convergence is uncertain. In this problem, these would be at the endpoints of the convergence interval, that is, at \(x = -1\) and \(x = 5\). Without additional information or a specific value for \(a_{k}\), we can't definitively say whether the series converges or diverges at these points.

Key Concepts

Radius of Convergence

The radius of convergence is a crucial aspect when dealing with power series. It determines the range of values for which the series converges - essentially the 'distance' from the center point within which the series behaves well. For a power series \[ \sum_{k=0}^{\infty} a_{k}(x-c)^{k} \]if the power series is given in the form centered at a point \(c\), the radius of convergence, \(R\), establishes this range. In our example, the series is centered at \(x = 2\) with a radius of convergence of \(3\). This means:

• For any value of \(x\) such that \(|x - 2| < 3\), the series converges.
• Values of \(x\) outside this boundary generally lead to divergence.

Getting the radius of convergence is often done using the ratio test or root test which involves an analysis of the series coefficients, \(a_k\).

Interval of Convergence

Once we have the radius of convergence, we can easily find the interval of convergence. This interval showcases all \(x\) values where the series converges absolutely. In our case, we start by applying the radius of the convergence:

• Center of series: \(x = 2\)
• Radius: \(3\)

This creates an interval from \((2 - 3)\) to \((2 + 3)\) which simplifies to \((-1, 5)\). It indicates the series will converge for any \(x\) value within this open interval. No analysis is needed beyond this unless one needs to check for endpoint behaviors.

Divergence of Series

It's important to determine when a series diverges because not all values will guarantee convergence. For a series, divergence means that the sum doesn't approach a finite value. Specifically, if \(|x - 2| > 3\), the distance exceeds the radius, causing the series to diverge. The divergence intervals are:

• To the left of \(-1\), meaning \(x < -1\).
• To the right of \(5\), meaning \(x > 5\).

In these cases, you can confidently assert divergence without needing additional information. This determination rests solely upon simple inequalities that stem from the radius of convergence.

Endpoints of Convergence

Endpoints in the interval of convergence can sometimes be tricky. While the interval \((-1, 5)\) guarantees convergence inside, the behavior at the points \(x = -1\) and \(x = 5\) is uncertain based solely on the radius. These endpoints are where we lack enough information to decisively conclude convergence or divergence:

• If \(x = -1\) or \(x = 5\), we enter a grey area. These are points where the radius of convergence test reaches its limits.
• The series may converge or diverge depending on specific series terms \(a_k\).

To resolve this ambiguity, one might conduct specific tests such as a direct test on those points or compare the series to known convergent or divergent series.