Problem 3 - Essential Calculus Early Transcendentals

Question

Suppose \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is known to be convergent. (a) If \(a_{n}>b_{n}\) for all \(n,\) what can you say about \(\Sigma a_{n} ?\) Why? (b) If \(a_{n}

Step-by-Step Solution

Answer

(a) \(\Sigma a_n\) diverges; (b) \(\Sigma a_n\) converges.

Step 1: Understanding Convergence and Divergence

For series with positive terms, convergence means the series sums to a finite value. If a series is convergent, then each term must get arbitrarily small but not necessarily zero. If a series is divergent, the series does not sum to a finite value.

Step 2: Analyzing Case (a): a_n > b_n for all n

Given that \(a_n > b_n\) for all \(n\), and \(\Sigma b_n\) is convergent, then \(\Sigma a_n\) must be divergent. This is because a convergent series \(\Sigma b_n\) with each \(b_n\) less than a corresponding \(a_n\) implies that \(\Sigma a_n\) is larger and thus cannot converge to a finite value.

Step 3: Analyzing Case (b): a_n < b_n for all n

For \(a_n < b_n\) and knowing \(\Sigma b_n\) is convergent, the series \(\Sigma a_n\) converges. This follows from the comparison test which states that if a series dominates another convergent series, the series with smaller terms is also convergent.

Key Concepts

Comparison Test

The Comparison Test is a fundamental tool used to determine whether a series is convergent or divergent. It involves comparing the terms of a given series with the terms of another series whose convergence is already known. This test can be quite intuitive:

• If you have two series, \( \Sigma a_n \) and \( \Sigma b_n \), both with positive terms, and \( 0 \leq a_n \leq b_n \) for all \( n \), then if \( \Sigma b_n \) is convergent, \( \Sigma a_n \) must also be convergent.
• Conversely, if \( a_n \geq b_n \) for all \( n \), and \( \Sigma b_n \) diverges, \( \Sigma a_n \) will also diverge.

This method works because a smaller series in terms of its elements can at most reach the sum of the larger one already known to converge. Similarly, if the larger series diverges, the smaller can never converge since it 'lifts' on the shoulders of larger terms.
Applying this test can simplify the process of analyzing series, especially when dealing with series involving complex terms.

Convergent Series

A convergent series is one where the sum of its infinite terms approaches a finite limit. This concept signifies that as you add more and more terms, the incremental additions become negligible, and you tend towards a specific value.
For a series \( \Sigma a_n \) to be convergent:

• The terms \( a_n \) must get smaller and eventually become close to zero.
• Mathematically, for any given small number \( \epsilon > 0 \), there exists a positive integer \( N \) such that for all \( n > N \), the partial sums \( S = a_1 + a_2 + ... + a_n \) is within \( \epsilon \) of the total sum \( S \).

In essence, this series never grows out of bounds but reaches a plateau where cumulative additions barely make a difference. Knowing this, when we find a series smaller or equal to a known convergent series, we can safely deduce the first series must converge as well, often using the Comparison Test.

Divergent Series

A divergent series is one where the sums of its terms don't approach a finite value. Instead, it continues to grow without bounds as more terms are added. This may occur for several reasons: the terms might not get small enough fast enough, or their cumulative sum could essentially be infinite.
When dealing with series \( \Sigma a_n \):

• Typically, if the terms \( a_n \) do not approach zero, the series will be divergent.
• Even if \( a_n \) approaches zero, this alone is not enough to ensure convergence. Sometimes more is going on with how terms add up than just individual term size.

If we use the Comparison Test, any series compared to a divergent one and found equal or larger will also be divergent. This principle helps identify divergence in many mathematical analyses, confirming that the comparison approach is crucial when evaluating series behavior.