Problem 66 - Thomas Calculus

Question

Show by example that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) may diverge even if \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge.

Step-by-Step Solution

Answer

Choose sequence \( a_n = \frac{1}{2^n} \) and \( b_n = (-1)^n \); then \( \sum_{n=1}^{\infty} a_n b_n = \sum_{n=1}^{\infty} \frac{(-1)^n}{2^n} \) diverges.

Step 1: Define Sequences

Let's define the sequences \( \{a_n\} \) and \( \{b_n\} \). Choose \( a_n = \frac{1}{2^n} \) and \( b_n = (-1)^n \). These sequences will help demonstrate that \( \sum_{n=1}^{\infty} a_{n} b_{n} \) may diverge.

Step 2: Confirm Convergence of \( \sum a_n \)

The series \( \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \frac{1}{2^n} \) is a geometric series with a common ratio \( r = \frac{1}{2} \), which is less than 1. Therefore, \( \sum_{n=1}^{\infty} a_n \) converges.

Step 3: Confirm Convergence of \( \sum b_n \)

Since \( b_n = (-1)^n \), consider the series \( \sum_{n=1}^{\infty} b_n \). It alternates between -1 and 1, so the partial sums are \( 0, -1, 0, -1, ... \). This series technically doesn't converge, but the terms \( b_n \) are bounded, leading us to focus instead on the product \( a_n b_n \) remaining bounded.

Step 4: Calculate \( a_n b_n \)

Compute the product for each term: \( a_n b_n = \frac{1}{2^n} \cdot (-1)^n = \frac{(-1)^n}{2^n} \). It is crucial to see how these terms interact in the series sum.

Step 5: Analyze the Series \( \sum a_n b_n \)

The series \( \sum_{n=1}^{\infty} a_n b_n = \sum_{n=1}^{\infty} \frac{(-1)^n}{2^n} \) is an alternating series. Despite \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) being bounded, this alternating series fluctuates indefinitely, indicating it does not converge.

Key Concepts

Alternating Series

An alternating series is one where the terms alternate in sign. This means the sequence of partial sums can jump back and forth. A classic example is a series where each term takes a form like

• \((-1)^n a_n\)

Here, the terms alternate between positive and negative as in \((-1)^n\).

The convergence of an alternating series can often be understood through the Alternating Series Test. According to this test, if the absolute value of the terms \(a_n\) is decreasing towards zero, then the alternating series converges.

In the specific exercise, the series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{2^n}\) is an alternating series, but it doesn't converge in the traditional sense because the series representing \(b_n\) itself was not convergent.

Geometric Series

A geometric series is a series where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio \(r\). A geometric series often looks like:

• \(a + ar + ar^2 + ar^3 + \ldots\)

Whether a geometric series converges or diverges mainly depends on the value of \(r\).

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

In the solution, the series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) is a geometric series with the common ratio \(r = \frac{1}{2}\). Because \(\frac{1}{2}\) is less than 1, this series converges.

Divergence of Series

A series diverges when the sequence of its partial sums doesn't approach a finite limit. Understanding divergence involves recognizing different causes behind why series don't settle to a number.

Some common reasons for divergence include:

• The terms of the series do not tend towards zero. This is a hint that their sum might not stabilize.
• The addition of infinitely many oscillating terms, similar to what happens in certain alternating series where partial sums perpetually bounce.

In this exercise, even though the series \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) and \(\sum_{n=1}^{\infty} (-1)^n\) are convergent in their respective ways, their multiplicative combination as \(\sum_{n=1}^{\infty} \frac{(-1)^n}{2^n}\) behaves chaotically and diverges.

In practice, when assessing the convergence or divergence of a product of series, it's essential to evaluate how the interaction between sequences affects the behavior of the entire sum.