Question

The possibilities listed in Theorem \(4.5 .2(\mathbf{c})\) for behavior of a power series at the endpoints of its interval of convergence do not include absolute convergence at one endpoint and conditional convergence or divergence at the other. Why can't these occur?

Short answer

In summary, the behavior of a power series at its endpoints cannot include absolute convergence at one endpoint and conditional convergence or divergence at the other because it contradicts Theorem 4.5.2(c). This theorem provides a list of possible behaviors at the endpoints of the interval of convergence, which does not include a scenario where one endpoint is absolutely convergent, and the other is conditionally convergent or divergent. This contradiction implies that such a scenario cannot occur within the interval of convergence for a power series.

Step-by-step solution

Recall Theorem 4.5.2(c)#

Theorem 4.5.2(c) states that for a power series, the possible behaviors at the endpoints of the interval of convergence include: 1. Both endpoints are absolutely convergent. 2. Both endpoints are conditionally convergent. 3. Both endpoints are divergent. 4. The power series converges at one endpoint and diverges at the other. Notice that there is no option which considers the case where a power series is absolutely convergent at one endpoint and conditionally convergent or divergent at the other endpoint. We will now discuss why this is the case.

Properties of Absolute Convergence, Conditional Convergence, and Divergence#

For a given power series, the following properties hold: 1. If a series is absolutely convergent, then it is also convergent. 2. A series is conditionally convergent if it is convergent but not absolutely convergent. 3. A series is divergent if it does not converge.

Explaining the Behavior of a Power Series at its Endpoints#

Let's consider a power series centered at \(x = 0\) with a radius of convergence \(R > 0\). Recall that the interval of convergence is \([-R, R]\). Suppose that the power series were absolutely convergent at \(x = R\) and conditionally convergent or divergent at \(x = -R\). By definition, an absolutely convergent series is convergent, so the series also converges at \(x = R\). Now, since the function is defined at the other endpoint \(x = -R\) and it is convergent (by conditional convergence) or divergent, it is either converging or diverging. However, this directly contradicts Theorem 4.5.2(c), which states that if a power series is convergent at one endpoint, it must also be convergent or divergent at the other. Therefore, it is not possible for a power series to have absolute convergence at one endpoint and conditional convergence or divergence at the other endpoint. This explains why Theorem 4.5.2(c) does not include this case in its list of possible behaviors for the endpoints of the interval of convergence.

Key concepts

Absolute Convergence

When you come across the term absolute convergence, it's important to understand its significance in relation to power series. Absolute convergence means that the sum of the absolute values of the series' terms converges. In more mathematical language, a series \( \sum a_n x^n \) is said to be absolutely convergent if \( \sum |a_n x^n| \) is convergent for a given value of \( x \).

This concept is essential because of a key property: if a series is absolutely convergent, it is guaranteed to be convergent as well. This has profound implications for a power series, especially when analyzing its behavior at the endpoints of the interval of convergence — it simply cannot be absolutely convergent at one endpoint and conditionally convergent or divergent at the other, as this would contradict the nature of absolute convergence.

Understanding absolute convergence is pivotal when you're attempting to determine the nature of a series because it lays the groundwork for assessing its overall convergent behaviors.

Conditional Convergence

On the flip side, we encounter conditional convergence. This occurs when a series converges, but it does not converge absolutely. It tells us that the convergence of the series is 'conditional' on the order of the terms. If you were to rearrange the terms, the series might not converge to the same sum, and it could even diverge.

An example to illustrate this is the alternating harmonic series, which converges conditionally but not absolutely. The distinction between absolute and conditional convergence is subtle but significant. In the context of power series, conditional convergence is possible at an endpoint, but as previously discussed, it can't coexist with absolute convergence at another endpoint within the same interval of convergence.

Interval of Convergence

Now let's focus on the interval of convergence. For a power series centered at a point \( c \), this interval represents all the values of \( x \) for which the series converges. It plays a vital role in understanding where a power series 'works' and where it doesn't.

Typically, this interval is symmetric around the center point \( c \) and is often described by \( c - R \) and \( c + R \) where \( R \) is the radius of convergence. Within this interval, the series converges; outside, it diverges. However, the tricky part is at the endpoints of this interval, where the behavior of the power series can vary — it might converge absolutely, converge conditionally, or not converge at all. The interval of convergence is crucial for knowing the boundaries within which a power series provides valid results.

Theorem 4.5.2(c)

To bring home these concepts, we look at Theorem 4.5.2(c). This theorem describes the possible behaviors of a power series at the endpoints of its interval of convergence. The options are neatly laid out: either both endpoints exhibit absolute convergence, conditional convergence, divergence, or there is a mix of convergence and divergence, but never a mix of absolute convergence and conditional convergence or divergence.

In essence, Theorem \(4.5.2(c)\) is a guideline for understanding endpoint behaviors of power series. It clarifies why you cannot have a power series absolutely converging at one endpoint and conditionally converging or diverging at the other. This theorem is your cheat sheet for predicting how a power series will behave at the edges of its realm — the interval of convergence.