Problem 46 - Calculus: Early Transcendentals

Question

If \( \sum a_n \) and \( \sum b_n \) are both convergent series with positive terms, is it true that \( \sum a_n b_n \) is also convergent?

Step-by-Step Solution

Answer

No, \( \sum a_n b_n \) is not necessarily convergent without additional conditions.

Step 1: Understanding Convergent Series

A convergent series is one where the sum of its terms approaches a finite limit as the number of terms goes to infinity. For the series \( \sum a_n \) and \( \sum b_n \) to be convergent, each of them must have partial sums which approach a finite value as more terms are added.

Step 2: Analyze the Product Series

We need to investigate the behavior of the series \( \sum a_n b_n \). For each pair of corresponding terms from the original series, we form a new sequence \( c_n = a_n b_n \). The question asks whether the sum \( \sum c_n = \sum a_n b_n \) is convergent.

Step 3: Assess the Convergence of the Product Series

Even though both series \( \sum a_n \) and \( \sum b_n \) are convergent, this does not guarantee the convergence of \( \sum a_n b_n \). A famous result, known as the Cauchy product, applies to absolutely convergent series and may not hold here because we only know the series have positive terms, not whether they are absolutely convergent.

Step 4: Consider a Counterexample

Consider \( a_n = \frac{1}{n} \) and \( b_n = \frac{1}{n} \). Both \( \sum a_n \) and \( \sum b_n \) are divergent since the harmonic series diverges, hence a counterexample with convergent series needs to involve other types of sequences. For instance, \( a_n = \frac{1}{n^p} \) and \( b_n = \frac{1}{n^q} \) with \( p, q > 1 \) are convergent individually, but \( a_n b_n = \frac{1}{n^{p+q}} \), which is convergent only for \( p+q > 1 \) under direct comparison to a \( p \)-series.

Step 5: Conclude Non-convergence in General

Since we can concoct sequences like \( a_n = \frac{1}{n} \) and \( b_n = \frac{1}{n^2} \) where \( \sum a_n \), \( \sum b_n \) converge, but \( \sum \frac{1}{n^3} \) converges instead, showing theoretically it's not guaranteed without specific conditions. Therefore in general situations, \( \sum a_n b_n \) is not necessarily convergent.

Key Concepts

convergent series with positive terms

A series is called convergent if its sum reaches a finite limit as more terms are added. One common way to determine convergence is through its partial sums. If these partial sums get closer and closer to a certain number, we say the series converges to that number. For instance, consider a series of positive terms, such as the geometric series. This series will converge if the absolute value of the common ratio is less than one.

Convergent series with positive terms give a sense of approximation towards stability in value. It's like filling a jar with water little by little, and eventually, you won't spill anymore as the jar reaches full capacity. This mathematical "fill level" is the finite limit of a convergent series.

Cauchy product

The Cauchy product is a method of multiplying two infinite series, producing a new series. Even if each of the original series is convergent with positive terms, the Cauchy product may not always result in a convergent series. This is because convergence behavior can drastically change when two series are multiplied together.

In simple terms, imagine you are mixing two stable substances. Even if each is stable on its own, the resulting mixture may not be. Similarly, with the Cauchy product, one famous result tells us that it only guarantees convergence if each of the original series is absolutely convergent. Absolute convergence is a stronger condition than just convergence of positive terms.

harmonic series

The harmonic series is one of the most well-known examples of divergent series. It is characterized by its terms being the reciprocals of natural numbers, such as \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots \). Despite its terms diminishing to zero, the harmonic series does not approach a finite limit as more terms are added, hence it diverges.

This series demonstrates an important lesson in the world of infinite series: not all series with terms that shrink to zero actually converge. The harmonic series reminds us that checking convergence goes beyond simply looking at individual terms.

p-series

The \( p \)-series is a general type of series defined as \( \sum \frac{1}{n^p} \) where \( p \) is a positive real number. The convergence of a \( p \)-series depends critically on the value of \( p \).

• If \( p > 1 \), the \( p \)-series converges. For example, \( \sum \frac{1}{n^2} \) is a convergent series.
• If \( p \leq 1 \), the \( p \)-series diverges. A common example is the harmonic series, \( \sum \frac{1}{n} \), where \( p = 1 \).

Understanding \( p \)-series is fundamental in determining whether specific infinite series will converge, particularly when they involve powers of natural numbers. This makes \( p \)-series a powerful tool in evaluating the behavior of sequences and their summation.