Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) both diverge, does it follow that \(\left\\{a_{n}+b_{n}\right\\}\) diverges?

Step-by-Step Solution

Answer

No, divergent sequences can be combined to form a convergent sequence.

Step 1: Understanding Divergence

The sequences \( \{a_n\} \) and \( \{b_n\} \) are given to be divergent. This means neither sequence converges to a specific limit. Divergence could imply they tend toward infinity, negative infinity, or oscillate without settling.

Step 2: Exploring Sum of Divergent Sequences

Consider what happens if we add two divergent sequences. For example, cumulative divergence could result in convergence (e.g., adding positively and negatively divergent sequences might result in a bounded and convergent sequence).

Step 3: Example Counterexample with Oscillating Sequences

Suppose \( \{a_n\} = (-1)^n \) and \( \{b_n\} = (-1)^{n+1} \). Both sequences diverge due to oscillation between -1 and 1. Now consider the sequence \( \{a_n + b_n\} \), which yields \(((-1)^n + (-1)^{n+1}) = 0\) for all \( n \). The sequence \( \{a_n + b_n\} \) is constant at 0, showing convergence.

Key Concepts

Divergent Sequences

Divergent sequences are fascinating. These are sequences where the terms do not settle down to a single number as you continue adding more terms. In other words, a divergent sequence does not approach a specific limit.

There are a few ways a sequence might diverge:

• It may increase or decrease without bound, heading towards infinity or negative infinity.
• It may oscillate, never settling into a steady pattern or limit.

Understanding divergent sequences is vital when considering the behavior of different mathematical expressions or real-world phenomena. As we saw in the exercise, both \( \{a_n\} \) and \( \{b_n\} \) are divergent, which means they do not converge to a single limit. However, divergence alone does not dictate what will happen when such sequences are summed with another divergent sequence.

Sequence Convergence

Sequence convergence is a condition where a sequence approaches a certain number as more terms are added. Convergent sequences tend to become more predictable and less erratic.

For a sequence \( \{a_n\} \) to be considered convergent, it must approach a number \( L \) such thatthe difference between \( a_n \) and \( L \) becomes extremely small as \( n \) becomes large.

In simple terms, convergent sequences "settle down" to a specific value.

• This behavior contrasts the unpredictability of divergent sequences.
• Mathematically, we say that \( \lim_{n \to \infty} a_n = L \).

The exercise showed us that even the sum of two divergent sequences can unexpectedly converge, like the pair of oscillating sequences \( \{a_n\} \) and \( \{b_n\} \) which individually diverge but whose sum remains constant at 0.

Oscillating Sequences

Oscillating sequences are a specific type of divergent sequence. These sequences alternate between values instead of trending towards infinity or negative infinity.

An oscillating sequence can best be exemplified by the sequence \( \{a_n\} = (-1)^n \), where the terms flip between -1 and 1 as \( n \) increases. This sequence never settles to one number and thus diverges.

• Oscillating sequences often appear in mathematical proofs and real-world phenomena, particularly those involving cyclic behavior.
• The regular pattern may seem manageable, yet it does not lead to a single limit.

In the exercise, we explored the fascinating case of adding two oscillating sequences. Each was diverging on its own, but their sum was surprisingly non-oscillating and constant at zero.This result highlighted how oscillating sequences can sometimes combine to form sequences with completely different characteristics.