Problem 42 - Calculus: Early Transcendentals

Question

Give an example of a pair of series \( \sum a _n \) and \( \sum b_n \) with positive terms where \( \lim_{n \to \infty} (a_n/b_n) = 0 \) and \( \sum b_n \) diverges, but \( \sum a_n \) converges. (Compare with Exercise 40.)

Step-by-Step Solution

Answer

Choose \( a_n = \frac{1}{n^2} \) and \( b_n = \frac{1}{n} \).

Step 1: Define the Series

Let's define the series we will work with. We choose the series \( \sum a_n = \sum \frac{1}{n^2} \), which is known to converge, and \( \sum b_n = \sum \frac{1}{n} \), which is known to diverge.

Step 2: Evaluate the Limit

Consider the sequence of terms \( a_n = \frac{1}{n^2} \) and \( b_n = \frac{1}{n} \). We need to calculate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{n^2}}{\frac{1}{n}} \).

Step 3: Simplify the Limit Expression

Simplify the expression for the limit: \( \frac{\frac{1}{n^2}}{\frac{1}{n}} = \frac{1}{n} \). So we have \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Step 4: Conclusion Based on Results

Since \( \lim_{n \to \infty} \frac{a_n}{b_n} = 0 \) and \( \sum b_n \) diverges while \( \sum a_n \) converges, we have found an example that satisfies the conditions of the problem.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool in determining the convergence or divergence of series. This test compares two series by looking at the behavior of their terms as they approach infinity. Here's how it works:

• Take two series with positive terms, say \( \sum a_n \) and \( \sum b_n \), and compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If this limit is a positive finite number, then both series either converge or diverge together.
• If the limit is zero and \( \sum b_n \) diverges, \( \sum a_n \) might still converge, like in our original exercise. This is an exception rather than a typical behavior.
• If the limit is infinite and \( \sum b_n \) converges, \( \sum a_n \) diverges. This is also an exceptional case.

Essentially, the test tells you that if \( \frac{a_n}{b_n} \) approaches an intuitive rate, it indicates they share the same convergence property, except when the limit equals zero or infinity, which needs personal verification.

Convergent Series

A convergent series is a series whose terms approach a limit, meaning their sum adds up to a finite number. Understanding convergent series can simplify many analysis problems, especially when dealing with infinite processes.

• A classic example is the series \( \sum \frac{1}{n^2} \). This series converges, as proven by tests like the p-series rule, which states if \( p > 1 \), the series \( \sum \frac{1}{n^p} \) converges.
• Convergence implies that, no matter how far you extend the sum, it gets closer and closer to a specific number, and never goes back out to infinity.
• This contrasts with what students may assume from the series' infinite nature. It is a crucial distinction in understanding infinite sequences and series.

The significance lies in knowing that with convergent series, errors can be controlled and managed, making them approachable and predictable within an infinite scope.

Divergent Series

In contrast, a divergent series does not settle into a finite sum. Instead, its total tends to infinity, or does not stabilize to a particular value. Grasping this concept is key to ensuring you can identify series that grow without bound.

• Consider the series \( \sum \frac{1}{n} \), also known as the harmonic series. Despite the terms getting smaller, the series diverges, meaning it grows indefinitely.
• A diverging series implies that continuity in growth between terms isn't at a rate sufficient to sum to a finite amount.
• Tests like the comparison test, ratio test, and, as in our exercise, the limit comparison test, help confirm divergence quickly and systematically.

Recognizing divergence ensures an understanding of limits, summations, and how certain mathematical models predict behavior over an infinite series.