Problem 43 - Calculus: One and Several Variables

Question

Let \(\sum a_{k} x^{k}\) be a series with radius of convergence \(r>0\) (a) Show that if the series is absolutely convergent at one endpoint of its interval of convergence, then it is absolutely convergent at the other endpoint. (b) Show that if the interval of convergence is \((-r . r)\) then the series is only conditionally convergent at \(r\)

Step-by-Step Solution

Answer

In short, we showed that if a power series is absolutely convergent at one endpoint of its interval of convergence, then it is also absolutely convergent at the other endpoint. Moreover, if the interval of convergence is (-r, r), then the series is conditionally convergent at r. We used the comparison test and ratio test to check for absolute convergence and the Alternating Series Test for convergence in our proofs.

Step 1: Part (a): Absolute convergence at both endpoints

Given an absolutely convergent series at one of the endpoints, without loss of generality, we can assume it is the left endpoint, \(-r\). Then, we have: \[\sum |a_{k}|r^{k}<\infty\] Now, consider the power series at the right endpoint \(r\). The corresponding series is: \[\sum |a_{k}|(-1)^kr^{k}\] To check the absolute convergence, we need to evaluate \(\sum |a_{k}|r^{k}\). However, we just showed that the left endpoint converges absolutely, so \(\sum |a_{k}|r^{k}<\infty\). So, the given series is absolutely convergent at both endpoints.

Step 2: Part (b): Conditional convergence at r

Assuming the interval of convergence is (-r, r), we must show that the series converges at r but is not absolutely convergent. Since the series converges within the interval (-r, r), we know that the limit of the general term goes to zero, i.e., \[ \lim_{k\to\infty}a_{k}r^{k}=0 \] Then, consider the alternating series at r: \[\sum a_{k}(-1)^kr^{k}\] We will use the Alternating Series Test for convergence. Since \(\lim_{k\to\infty}a_{k}r^{k}=0\), we have: \[ \lim_{k\to\infty}a_{k}(-1)^kr^{k}=0 \] Now we need to show that the sequence is decreasing. We know that for any convergent series the limit of the sequence as k goes to infinity is 0: \[ \lim_{k\to\infty}a_{k}=0 \] Thus, as k increases, the terms \(a_{k}\) become smaller. Since the series converges within the interval, we know the terms are decreasing. From the Alternating Series Test, the series converges at r. However, it's not absolutely convergent at r because the radius of convergence is precisely r. If it were absolutely convergent at r, the radius of convergence would be greater than r, violating the given scenario. Therefore, the series is conditionally convergent at r, as desired.

Key Concepts

Radius of Convergence

The radius of convergence is a crucial concept in understanding power series. This term refers to the distance from the center of the series where the series converges. In simpler words, if you picture the series centered at zero, it's the radius within which the series will definitely converge. This is often represented as \( r \). If you push beyond this distance, the series isn't guaranteed to converge.

The radius of convergence can be found using formulas such as the ratio test or the root test. Specifically, for the power series \( \sum a_k x^k \), if using the ratio test, you look at \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). The series will converge when this limit is less than 1. That limit helps you calculate \( r \). A larger radius implies that the series converges for a broader range of \( x \) values. It’s a powerful way to understand where your series works safely.

Absolute Convergence

Absolute convergence is when the series \( \sum |a_k x^k| \) converges. This means if you take the absolute value of each term and add them up, the sum comes to a finite number. If a series converges absolutely, it means it is super stable. No matter how you shuffle the terms around, the series will still converge to the same value.

In the exercise, absolute convergence at one endpoint implies it at the other. When dealing with endpoints of the interval \( (-r, r) \), showing absolute convergence is a way of confirming the convergence stability. So, if \( \sum |a_{k} r^{k}| < \infty \) holds at one endpoint, it simply means that, since this endpoint behaves, the other must do as well. It’s like a double-check of convergence physics for power series.

Conditional Convergence

Conditional convergence occurs when a series converges, but it does not converge absolutely. This convergence condition is more delicate. If you take the absolute values of every term and the series diverges, yet the series without the absolute values converges, it's conditionally convergent.

In the problem, if a series converges conditionally at \( r \), then while the series itself converges, becoming too enthusiastic by taking the absolute value would make it blow up (diverge). This was shown in the exercise through the presence of a convergent alternating series at \( r \), yet highlighting that an absolute approach wouldn't work there.

Interval of Convergence

The interval of convergence is the range of \( x \) values for which the series \( \sum a_k x^k \) converges. Identifying this interval is essential in analyzing the behavior of your series.

You can view it as the span of \( x \) values from the center of the series (usually 0) within the radius of convergence and possibly include endpoints. This interval plays a pivotal role because it tells you exactly between which two solid points your power series behaves well.

For example, if the interval of convergence is \((-r, r)\) not including endpoints, the series is safe within but risky on points like \( -r \) or \( r \) without testing. Thorough testing for absolute or conditional convergence at these boundaries is crucial, often using series tests.

Alternating Series Test

The alternating series test is a handy tool to check for convergence in series with alternating positive and negative terms. Think about a series like this: \( \sum (-1)^k a_k x^k \). This test asserts that such a series will converge if two main conditions are met:

• The absolute values of the terms \( a_k \) must be decreasing; \( |a_{k+1}| \leq |a_k| \).
• The limit of these terms must approach zero as \( k \) grows;
  that means \( \lim_{k \to \infty} a_k = 0 \).

The alternating series test was applied in the exercise for making sure the series converges at \( r \). Confirming that each condition was held proved the convergence for a conditionally convergent scenario, making sure it behaves but not absolutely because the series is only stable, not absolutely stable there.