Question

Prove that the Cauchy product of two absolutely convergent series converges absolutely.

Short answer

Question: Prove that the Cauchy product of two absolutely convergent series converges absolutely. Answer: Using the definition of the Cauchy product and the properties of absolutely convergent series, we can show that the sum of the absolute values of the Cauchy product terms converges. By applying the triangle inequality property and comparing the result with the product of the two absolutely convergent series, we can conclude that the Cauchy product converges absolutely.

Step-by-step solution

Definition of Cauchy Product

Given two series, ∑a_n and ∑b_n, their Cauchy product is defined as a new series ∑c_n, where: c_n = a_0 * b_n + a_1 * b_(n-1) + ... + a_(n-1) * b_1 + a_n * b_0.

Define Absolutely Convergent Series

A series is said to be absolutely convergent if the series of its absolute values converges, that is, ∑|a_n| and ∑|b_n| are convergent.

Apply Triangle Inequality Property

To show that the Cauchy product converges absolutely, we need to prove that the sum of the absolute values of the Cauchy product terms converges: ∑|c_n| = ∑|a_0 * b_n + a_1 * b_(n-1) + ... + a_(n-1) * b_1 + a_n * b_0|. Using the triangle inequality property, we have: |c_n| ≤ |a_0 * b_n| + |a_1 * b_(n-1)| + ... + |a_(n-1) * b_1| + |a_n * b_0|.

Apply Absolute Convergence Property

Since ∑|a_n| and ∑|b_n| are absolutely convergent, there exist constants A and B such that: ∑|a_n| = A and ∑|b_n| = B.

Product of Absolutely Convergent Series

Now, we consider the product of these two absolutely convergent series, ∑|a_n| * ∑|b_n|: A * B = (∑|a_n|) * (∑|b_n|). Using properties of series, we can rewrite this as: A * B = ∑(∑|a_k * b_n−k|) for k=0 to n. Then, by comparing with the previous inequality: A * B ≥ ∑|c_n|

Prove Cauchy Product Convergence

Since A * B is a finite number, and |c_n| is always less than or equal to A * B, we can conclude from the comparison test that ∑|c_n| converges. Therefore, the Cauchy product of two absolutely convergent series converges absolutely.

Key concepts

Absolute Convergence

Absolute convergence is an important concept in the world of series. It pertains to the behavior of series when considering their absolute values.
For a series \( \sum a_n \) to be absolutely convergent, the series formed by taking the absolute values of its terms, \( \sum |a_n| \), must converge. This means the total sum of the absolute values is finite.
This property is quite powerful because if a series is absolutely convergent, it guarantees that the series is also convergent in the usual sense (without taking absolute values). This ensures stability in many mathematical operations, such as rearranging the order of terms, which wouldn't usually affect the outcome for an absolutely convergent series.

• Absolute convergence implies convergence.
• Helps in rearranging series terms without altering the limit.
• Ensures finite sum of absolute values.

In the exercise, absolute convergence is central to proving that a Cauchy product also converges absolutely when formed by two absolutely convergent series.

Series Convergence

Series convergence is a fundamental concept that describes how the sum of an infinite series behaves. A series is said to converge if the sequence of its partial sums approaches a specific number; in other words, it has a finite limit.
There are many tests that can determine whether a specific series converges or not. Common tests include the Ratio Test, the Root Test, and the Comparison Test. Each of these tests examines a different aspect of the series to determine its behavior over time.
In the problem, series convergence is examined in the context of absolute convergence. The goal is to establish that the Cauchy product of two absolutely convergent series is also absolutely convergent. This involves showing that the sequence of partial sums of the Cauchy product approaches a finite limit, harnessing the property that both original series are absolutely convergent.

Triangle Inequality

The triangle inequality is a mathematical rule that helps in understanding the sum of two or more vectors or numbers. It states that the absolute value of a sum is less than or equal to the sum of the absolute values of the individual components.
In mathematical notation, for any numbers or vectors \( x \) and \( y \), the inequality is expressed as:
\( |x + y| \leq |x| + |y| \).
This property is very useful when dealing with series, especially when you are analyzing the absolute convergence of a series. It helps manage the sum of a series by considering the individual sum of absolute values instead.

• Provides an upper bound for absolute values of sums.
• Facilitates understanding of convergence by comparing absolute values.

In the solution, the triangular inequality is applied to manage and derive upper bounds for the absolute values of the terms in the Cauchy product, ensuring that these are bounded by the series' known convergent sums.