Problem 70 - Thomas Calculus

Question

Outline of the proof of the Rearrangement Theorem (Theorem 17\()\) $$\begin{array}{l}{\text { a. Let } \epsilon \text { be a positive real number, let } L=\sum_{n=1}^{\infty} a_{n}, \text { and let }} \\\ {s_{k}=\sum_{n=1}^{k} a_{n} \text { . Show that for some index } N_{1} \text { and for some }} \\ {\quad \text { index } N_{2} \geq N_{1}}\end{array}$$ $$\sum_{n=N_{1}}^{\infty}\left|a_{n}\right|<\frac{\epsilon}{2} \quad \text { and } \quad\left|s_{N_{2}}-L\right|<\frac{\epsilon}{2}$$ $$\begin{array}{l}{\text { Since all the terms } a_{1}, a_{2}, \ldots, a_{N_{2}} \text { appear somewhere in the }} \\ {\text { sequence }\left\\{b_{n}\right\\}, \text { there is an index } N_{3} \geq N_{2} \text { such that if }} \\ {n \geq N_{3}, \text { then }\left(\sum_{k=1}^{n} b_{k}\right)-s_{N_{2}} \text { is at most a sum of terms } a_{m}} \\ {\text { with } m \geq N_{1} . \text { Therefore, if } n \geq N_{3}}\end{array}$$ $$\begin{aligned}\left|\sum_{k=1}^{n} b_{k}-L\right| & \leq\left|\sum_{k=1}^{n} b_{k}-s_{N_{2}}\right|+\left|s_{N_{2}}-L\right| \\ & \leq \sum_{k=N_{1}}^{\infty}\left|a_{k}\right|+\left|s_{N_{2}}-L\right|<\epsilon \end{aligned}$$ $$\begin{array}{l}{\text { b. The argument in part (a) shows that if } \sum_{n=1}^{\infty} a_{n} \text { converges }} \\ {\text { absolutely then } \sum_{n=1}^{\infty} b_{n} \text { converges and } \sum_{n=1}^{\infty} b_{n}=\sum_{n=1}^{\infty} a_{n}} \\ {\text { Now show that because } \sum_{n=1}^{\infty}\left|a_{n}\right| \text { converges, } \sum_{n=1}^{\infty}\left|b_{n}\right|} \\ {\text { converges to } \sum_{n=1}^{\infty}\left|a_{n}\right|}\end{array}$$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) converge to the same limit, and their absolute values also converge equally.

Step 1: Establish Existence of N1 and N2

Given that the series \( \sum_{n=1}^{\infty} a_n = L \) converges, for any small positive number \( \epsilon \), there exists a certain index \( N_1 \) such that the series' tail \( \sum_{n=N_1}^{\infty} |a_n| < \frac{\epsilon}{2} \). Similarly, there must be an index \( N_2 \geq N_1 \) such that the partial sum \( s_{N_2} = \sum_{n=1}^{N_2} a_n \) satisfies \( |s_{N_2} - L| < \frac{\epsilon}{2} \). These ensure that the majority of the sum's magnitude is captured by the first \( N_2 \) terms, and the tail's effect is minimal.

Step 2: Rearrangement and Additional Index N3

Because each \( a_{n} \) appears somewhere in a rearrangement sequence \( \{ b_n \} \), there exists an index \( N_3 \geq N_2 \) such that for all \( n \geq N_3 \), the sum \( \sum_{k=1}^{n} b_k - s_{N_2} \) involves only the terms \( a_m \) with \( m \geq N_1 \). Thus the additional terms in the partial sum after \( N_2 \) are merely a rearrangement of \( a_m \) for \( m \geq N_1 \), whose total absolute value is already considered small (less than \( \frac{\epsilon}{2} \)).

Step 3: Bound the Rearranged Series' Deviation from Limit L

For any \( n \geq N_3 \), calculate the deviation: \( \left|\sum_{k=1}^{n} b_k - L\right| \leq \left|\sum_{k=1}^{n} b_k - s_{N_2}\right| + \left|s_{N_2} - L\right| \). Since we've arranged \( b_k \) such that the sum only includes terms \( a_m \) with \( m \geq N_1 \), \( \left|\sum_{k=1}^{n} b_k - s_{N_2}\right| \leq \sum_{k=N_1}^{\infty}|a_k| < \frac{\epsilon}{2} \). Therefore, the overall deviation is bounded by \( \epsilon \) since \( \left|s_{N_2} - L\right| < \frac{\epsilon}{2} \).

Step 4: Concluding Convergence and Equality of Series

Part (a) ensures that if \( \sum_{n=1}^{\infty} a_n \) converges absolutely, then \( \sum_{n=1}^{\infty} b_n \) also converges and their limits are equal. Part (b) extends this conclusion to show that because the absolute convergence \( \sum_{n=1}^{\infty} |a_n| \) ensures \( \sum_{n=1}^{\infty} |b_n| \) converges, the sum of absolute values also remains equal before and after rearrangement.

Key Concepts

Absolute Convergence

In mathematics, the concept of absolute convergence is fundamental when dealing with infinite series. A series \( \sum_{n=1}^{\infty} a_n \) is said to converges absolutely if the series of its absolute values \( \sum_{n=1}^{\infty} |a_n| \) also converges. This is a stronger condition than simple convergence, which only requires that \( \sum_{n=1}^{\infty} a_n \) converges as an infinite sequence of partial sums.

Absolute convergence has important implications:

• If a series is absolutely convergent, it implies that rearranging the terms of the series will also result in a convergent series.
• Furthermore, the original series and any rearranged series all converges to the same sum.
• This property makes it easier and more flexible to manipulate the series without affecting its convergence or sum.

Series Convergence

Series convergence refers to the condition under which the sum of an infinite sequence, called a series, approaches a finite limit as the number of terms increases indefinitely. When discussing series convergence, two main types are usually considered: conditional convergence and absolute convergence. Understanding these distinctions is key to grasping the nature of infinite series:

• Conditional Convergence: When a series \( \sum_{n=1}^{\infty} a_n \) converges, but \( \sum_{n=1}^{\infty} |a_n| \) does not converge, the series is conditionally convergent. Such series can be quite sensitive to the order of their terms.
• Absolute Convergence: As mentioned earlier, if \( \sum_{n=1}^{\infty} |a_n| \) converges, then \( \sum_{n=1}^{\infty} a_n \) converges absolutely. This implies more stability in the convergence behaviors across different term arrangements.
• Convergence can be tested using various mathematical tests such as the Comparison Test, Ratio Test, and Integral Test among others.

Rearrangement of Series

The rearrangement of a series involves changing the order of its terms. For a series with terms \( \{a_n\} \), a rearrangement would be a series with terms \( \{b_n\} \), where each \( b_n \) is one of the \( a_n \) values. The Rearrangement Theorem concerns the effects of rearranging the terms of an infinite series. Here are the main points related to rearranging series:

• If a series is absolutely convergent, any rearrangement of its terms will also result in a series that converges to the same sum. This happens because absolute convergence ensures that the total "amount" or magnitude in a series doesn't depend on the order of addition.
• For conditionally convergent series, however, rearranging terms can dramatically alter the sum or even lead to divergence. This sensitivity is a key characteristic that distinguishes conditional convergence from absolute convergence.
• The theorem highlights why absolute convergence is a more desirable property when working with series, as it guarantees consistency and reliability in results, regardless of how terms are arranged.