Problem 72 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose \(\lim _{k \rightarrow \infty} \sqrt[k]{\left|a_{k}\right|}=\frac{1}{3}\). (a) What is the radius of convergence of \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\) ? (b) For each value of \(x\) listed below, determine whether the series converges absolutely, converges conditionally, or diverges. i. \(x=0\) ii. \(x=-3\) iii. \(x=-1.5\) iv. \(x=5\)

Step-by-Step Solution

Answer

The radius of convergence R is 3. The series converges absolutely for x = 0 and x = -1.5. The series diverges for x = -3 and x = 5.

Step 1: Find the radius of convergence

Using the root test, by the given limit may be written as \(|x-1|/R = 1/3\) where \(R\) is the radius of convergence.When the limits for convergence are solved, we obtain the radius \(R = 3\).

Step 2: Check convergence for x = 0

When \(x = 0\) in the series \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\), the absolute value of \(x - 1\) becomes 1 which as per the root test, is within the bounds of convergence. So, the series converges absolutely for \(x = 0\).

Step 3: Check convergence for x = -3

When \(x = -3\) in the series \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\), we get \(|-3 - 1| = 4 > 3\) (3 being our calculated radius). This is outside the region of convergence, so the series diverges at \(x = -3\).

Step 4: Check convergence for x = -1.5

When \(x = -1.5\) in the series \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\), we get \(|-1.5 - 1| = 2.5 < 3\). This is inside the region of convergence, so the series converges absolutely.

Step 5: Check convergence for x = 5

When \(x = 5\) in the series \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\), we get \(|5 - 1| = 4 > 3\). This is outside the region of convergence, so the series diverges.

Key Concepts

Root Test

The root test is a method used to determine the convergence or divergence of an infinite series. Given a series \( \sum_{k=1}^{\infty} a_k \), the root test involves taking the limit of the \(k\)-th root of the absolute value of the \(k\)-th term, that is \( \lim_{k \rightarrow \infty} \sqrt[k]{|a_k|} \).

If the limit is less than 1, the series converges absolutely; if the limit is greater than 1, the series diverges; and if the limit equals 1, the test is inconclusive. In the exercise provided, the result of the root test \( \lim_{k \rightarrow \infty} \sqrt[k]{|a_k|} = \frac{1}{3} \) suggests that the series converges absolutely.

Series Convergence

Series convergence refers to whether the summation of an infinite sequence of terms approaches a finite value as more and more terms are added. A series \( \sum_{k=1}^{\infty} a_k \) converges if the partial sums of the series form a sequence that converges to a specific number or expression.

In the context of the exercise, for values of \(x\) within a certain range, the series will converge to a finite sum. Identifying this range depends on the convergence criteria, such as the radius of convergence found using the root test.

Absolute Convergence

A series is said to converge absolutely if the series formed by taking the absolute value of each term also converges. In mathematical terms, if \( \sum_{k=1}^{\infty} |a_k| \) converges, then the original series \( \sum_{k=1}^{\infty} a_k \) also converges, and it does so absolutely.

This concept ensures that rearranging the terms of the series \( \sum_{k=1}^{\infty} a_k \) does not affect the sum to which it converges. The exercise demonstrated this when checking for specific values of \(x\), such as \(x=0\) and \(x=-1.5\), where the absolute value was within the radius of convergence, leading to absolute convergence.

Conditional Convergence

Conditional convergence occurs when a series \( \sum_{k=1}^{\infty} a_k \) converges, but the series of absolute values \( \sum_{k=1}^{\infty} |a_k| \) does not converge. The original series will then only converge when the terms are added in a particular order.

In many cases, if a series converges conditionally, it is sensitive to the arrangement of the terms. This concept is not represented in the original problem given but is important in understanding the nature of series that do not pass the test for absolute convergence.

Divergence of Series

The divergence of a series occurs when the series does not converge to a finite value. For example, as seen in the original exercise, when checking for \(x=-3\) and \(x=5\), the absolute value of \(x - 1\) exceeded the radius of convergence.

This means that the partial sums of the series grow without bound and therefore, the series diverges. Divergence implies that there is no finite sum or meaningful total that can be ascribed to the series.