Problem 65 - Thomas Calculus

Question

Show that if \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge absolutely, then so do the following. $$\begin{array}{ll}{\text { a. }} & {\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)} & {\text { b. } \sum_{n=1}^{\infty}\left(a_{n}-b_{n}\right)} \\ {\text { c. }} & {\sum_{n=1}^{\infty} k a_{n}(k \text { any number })}\end{array}$$

Step-by-Step Solution

Answer

All three series \( \sum_{n=1}^{\infty} (a_n + b_n) \), \( \sum_{n=1}^{\infty} (a_n - b_n) \), and \( \sum_{n=1}^{\infty} k a_n \) converge absolutely.

Step 1: Understanding Absolute Convergence

If \( \sum_{n=1}^{\infty} a_n \) converges absolutely, then \( \sum_{n=1}^{\infty} |a_n| \) converges. Similarly, if \( \sum_{n=1}^{\infty} b_n \) converges absolutely, then \( \sum_{n=1}^{\infty} |b_n| \) converges.

Step 2: Adding Absolutely Convergent Series (Part a)

Consider the series \( \sum_{n=1}^{\infty} (a_n + b_n) \). Since absolute convergence of \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) implies \( \sum_{n=1}^{\infty} |a_n| \) and \( \sum_{n=1}^{\infty} |b_n| \) converge, then:\[ \sum_{n=1}^{\infty} |a_n + b_n| \leq \sum_{n=1}^{\infty} (|a_n| + |b_n|) = \sum_{n=1}^{\infty} |a_n| + \sum_{n=1}^{\infty} |b_n| \]This sum of convergent series is also convergent, hence \( \sum_{n=1}^{\infty} (a_n + b_n) \) converges absolutely by the comparison test.

Step 3: Subtracting Absolutely Convergent Series (Part b)

Consider the series \( \sum_{n=1}^{\infty} (a_n - b_n) \). Rewriting this, we have \( a_n - b_n = a_n + (-b_n) \).\Since \( \sum_{n=1}^{\infty} b_n \) converges absolutely, \( \sum_{n=1}^{\infty} |b_n| \) converges, and similarly \( \sum_{n=1}^{\infty} |-b_n| = \sum_{n=1}^{\infty} |b_n| \). Then:\[ \sum_{n=1}^{\infty} |a_n - b_n| = \sum_{n=1}^{\infty} |a_n + (-b_n)| \leq \sum_{n=1}^{\infty} (|a_n| + |-b_n|) = \sum_{n=1}^{\infty} (|a_n| + |b_n|) \]This, being a sum of convergent series, converges, meaning \( \sum_{n=1}^{\infty} (a_n - b_n) \) converges absolutely.

Step 4: Multiplying by a Constant (Part c)

Consider the series \( \sum_{n=1}^{\infty} k a_n \) where \( k \) is a constant. For absolute convergence:\[ \sum_{n=1}^{\infty} |k a_n| = |k| \sum_{n=1}^{\infty} |a_n| \]Since \( \sum_{n=1}^{\infty} |a_n| \) converges and \( |k| \) is a constant multiplier, \( |k| \sum_{n=1}^{\infty} |a_n| \) also converges. Thus, \( \sum_{n=1}^{\infty} k a_n \) converges absolutely.

Key Concepts

Series Convergence

A series convergence refers to whether the sum of an infinite series approaches a finite value as more and more terms are added. When dealing with infinite series such as \( \sum_{n=1}^{\infty} a_n \), we are often interested in whether this series converges to some number or not. The crux here is understanding the nature of the terms being added.

To determine whether a series converges, there are several tests that can be used. These tests often depend on the relationship between the terms of the series and their behavior as \( n \) tends towards infinity.

• If the sum reaches a specific value, the series converges.
• If the sum grows indefinitely, the series diverges.

In our problem, each series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) converges absolutely, implying that the sums of their absolute values also converge. Absolute convergence means not only does the series converge, but it does so even when considering all terms as positive by taking their absolute values.

Comparison Test

The comparison test is a handy method to determine the convergence of a series by comparing it to another series whose convergence properties are already known. The idea is straightforward: if you can establish that the terms of your series are always smaller or larger than the terms of a known series, conclusions can be drawn about its convergence.

Here's how it works:

• If \( 0 \leq a_n \leq b_n \) for all \( n \) and \( \sum_{n=1}^{\infty} b_n \) converges, then \( \sum_{n=1}^{\infty} a_n \) also converges.
• If \( a_n \geq b_n \geq 0 \) for all \( n \) and \( \sum_{n=1}^{\infty} b_n \) diverges, then \( \sum_{n=1}^{\infty} a_n \) diverges as well.

The comparison test supports our conclusions in the given problem by showing that adding, subtracting, or multiplying by constants of absolutely convergent series results in series that also converge absolutely. Particularly, for sums like \( (a_n + b_n) \) and \((a_n - b_n)\), knowing \( |a_n| + |b_n| \) converges is key.

Infinite Series

Infinite series are a fundamental part of calculus and analysis, comprising the sum of an infinite list of numbers. Understanding infinite series involves grasping concepts like convergence, divergence, and the behavior of sequences.

In mathematics, infinite series are often written as \( \sum_{n=1}^{\infty} a_n \). These series can behave in various ways:

• A convergent series adds up to a finite value, such as in geometric series where the sum has a limit imparting a stopping point as \( n \to \infty \).
• A divergent series does not settle at a limit and can continue to grow or oscillate indefinitely.

For instance, in our problem, the absolute convergence of \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) ensures that operations such as addition and subtraction of series maintain convergence. Working with infinite series, the challenge often dissipates to verifying the behavior of its terms and applying known tests, such as the comparison test, to confirm convergence properties.