Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that a conditionally convergent series can be rearranged so as to diverge.

Step-by-Step Solution

Answer

A conditionally convergent series can be rearranged to diverge using Riemann's Rearrangement Theorem.

Step 1: Understanding Conditional Convergence

A series \( \sum a_n \) is conditionally convergent if it converges, but the series of absolute values \( \sum |a_n| \) diverges. This means the positive and negative terms in the series cancel out just enough for the series to converge.

Step 2: Consider the Alternating Harmonic Series

The alternating harmonic series \( \sum (-1)^{n+1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \) is a classic example of a conditionally convergent series.

Step 3: Rearranging the Series

We rearrange the series by collecting two positive terms for every negative term in a way that alters the convergence. First, pick one positive term: \( 1 \). Then select enough negative terms to make the partial sum negative. Repeat this procedure indefinitely.

Step 4: Showing the Series Diverges

Whenever you keep adding two positive terms every large set of negative terms, the positive partial sums grow arbitrarily large, ensuring that the series diverges. Mathematically, you end up with increasing partial sums, demonstrating divergence.

Step 5: Reference to Riemann's Rearrangement Theorem

According to Riemann’s Rearrangement Theorem, any conditionally convergent series can be rearranged to converge to any given sum, or even to diverge. This highlights the instability of conditionally convergent series when rearranged.

Key Concepts

Riemann's Rearrangement Theorem

In the world of mathematics, series play a vital role in analysis. A fascinating aspect is Riemann's Rearrangement Theorem, which speaks about conditionally convergent series. These series, when rearranged, have an intriguing property. They can be manipulated to yield virtually any result you desire:

• Converge to any chosen value.
• Diverge to infinity or negative infinity.

The theorem highlights the flexibility and unpredictability of rearranging terms in these series. In simple terms, the way you order the terms can drastically change the series' behavior. This happens because conditionally convergent series balance between negative and positive terms in such a precise way that a little adjustment can cause huge changes. Essentially, Riemann's theorem showcases an unusual sensitivity present within these series, emphasizing the importance of order in mathematical operations.

Alternating Harmonic Series

An example to illustrate the idea of conditional convergence is the Alternating Harmonic Series. Represented as \(\sum (-1)^{n+1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots\), it alternates signs—positive, negative, positive, and so forth.
This specific sequence of numbers converges, meaning its total sum approaches a specific value. However, if we take the absolute values for comparison \(\sum \left| (-1)^{n+1} \frac{1}{n} \right| \), this sequence does not converge—it goes to infinity.
The alternating pattern ensures the positive and negative terms cancel each other out significantly but not entirely. Think of it like a delicate balancing act where the terms alternately pull the total in different directions, keeping it from shooting off to infinity.

Series Divergence

Series divergence occurs when the sum of the terms in a series does not approach a particular limit as the number of terms goes up indefinitely. It means adding up infinitely many terms leads to the series increasing more and more without ever settling down to a specific number.
In the context of conditionally convergent series, rearranging the terms can lead to divergence. By altering the order of terms, such as in Riemann’s method, it is possible to make the series grow indefinitely instead of converging. This manipulation causes the partial sums of the series to become increasingly large, and they do not tend to any fixed value.
Understanding the distinction between convergence and divergence is key. It highlights how adding terms in a different order changes the overall outcome drastically, making clear why order matters in infinite series.