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, \( \{a_n + b_n\} \) can converge if \( \{a_n\} \) and \( \{b_n\} \) diverge oppositely.

Step 1: Understanding Divergence

A sequence \( \{a_n\} \) is said to diverge if it does not approach any finite limit as \( n \to \infty \). Similarly, \( \{b_n\} \) diverges if it also lacks a finite limit. The question asks if their sum \( \{a_n + b_n\} \) must also lack a limit.

Step 2: Consider Convergence at Infinity

If both \( \{a_n\} \) and \( \{b_n\} \) approach \( +\infty \) or \( -\infty \), their sum will also diverge to the same infinity. However, they could diverge in a way that their terms have opposing trends (e.g., one tends to \(+\infty\) and the other to \(-\infty\)).

Step 3: Constructing a Counterexample

Consider sequences \( a_n = n \) and \( b_n = -n \). Both \( \{a_n\}\) and \( \{b_n\} \) diverge individually as \( n \to \infty \) and \( n \to -\infty \), respectively.

Step 4: Evaluate their Sum

Calculate \( a_n + b_n = n - n = 0 \). Hence, the sequence \( \{a_n + b_n\} \) becomes a constant sequence \( 0, 0, 0, \ldots \), which converges.

Step 5: Conclusion Based on Example

The sequence \( \{a_n + b_n\} \) converging shows that the sum of two divergent sequences does not necessarily diverge if their divergent behavior cancels out. Thus, it does not follow that \( \{a_n + b_n\} \) must diverge just because \( \{a_n\} \) and \( \{b_n\} \) do.

Key Concepts

Sequence Convergence

In mathematics, understanding sequence convergence is vital, especially when dealing with infinite nature. A sequence is simply an ordered list of numbers, like \( \{ a_n \} \). When we talk about convergence, we mean that no matter how far you go along the sequence, the numbers are getting closer and closer to a specific value. This value is known as the 'limit'. For instance, in a sequence like \( \{ 1, \frac{1}{2}, \frac{1}{3}, \ldots \} \, \) which approaches zero, we say it converges to zero as \( n o \infty \, \) which signifies infinite terms.

• Convergence is about approaching a finite limit.
• If a sequence stabilizes at a certain value, it converges.
• Convergent sequences are predictable, unlike divergent ones.

Conversely, if a sequence doesn’t settle at any number, it is considered divergent. In observing sequence convergence in exercises, always search for a pattern or consistency that leads to a specific number.

Limits of Sequences

The limit of a sequence \( \{ a_n \} \) is the value that \( a_n \) gets closer to as \( n \) becomes very large. This limit defines the convergence of the sequence. When we write \( \lim_{{n \to \infty}} a_n = L \, \) we mean as \( n\to\infty\) (goes to infinity), the sequence values approach the number \( L \.\) It can also be the case where \( L \) is \( +\infty \) or \( -\infty \).

• Not all sequences have a limit. Only convergent sequences reach a specific \( L \).
• In sequences where \( L \) is finite, we say the sequence converges.
• If no such finite \( L \) exists, the sequence diverges.

For example, if \( a_n = \frac{1}{n} \, \) then \( \lim_{{n \to \infty}} \frac{1}{n} = 0 \.\) As \( n \) becomes infinitely large, the terms of the sequence become infinitely close to 0, confirming its convergence.

Infinite Series

Infinite series are a bit different from sequences. Instead of a list of numbers, an infinite series is the sum of a sequence. It's often written as \( \sum_{n=1}^\infty a_n \), which means you'll add up all terms from \( a_1 \) to the point where \( n \) heads to infinity. The convergence of an infinite series depends on the sums of its terms becoming close to a particular number, called the sum of the series.

• If the partial sums approach a specific number, the series converges.
• If the sums do not settle, the series diverges.
• Divergent series spiral away, not stabilizing to a numeral value.

Recognizing the nature of a series as either convergent or divergent is essential in calculus and to solve real-world problems. For example, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is known to diverge, meaning it unsteadily accumulates more as terms go to infinity without ever stabilizing.