Problem 24 - Precalculus: Mathematical for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{(-1)^{n}}{n}$$

Step-by-Step Solution

Answer

The sequence is convergent with a limit of 0.

Step 1: Understand the Nature of the Sequence

The sequence given is \( a_n = \frac{(-1)^n}{n} \). This is an alternating sequence, meaning each subsequent term alternates between positive and negative due to \((-1)^n\). As \(n\) increases, the value of \(\frac{1}{n}\) gets smaller.

Step 2: Analyze the Behavior of Terms

When \(n\) is odd, \((-1)^n\) becomes \(-1\), causing \(a_n\) to be negative, such as \(-1/n\). When \(n\) is even, the term is positive, such as \(1/n\). Thus, the general behavior of the terms alternates signs and approaches zero as \(n\) increases.

Step 3: Check Convergence Using Limit Definition

To determine convergence, we find \(\lim_{{n \to \infty}} a_n\). Since \(a_n = \frac{(-1)^n}{n}\), observe that as \(n\) approaches infinity, \(\frac{1}{n} \to 0\). Consequently, the terms of the sequence also approach zero regardless of the sign alterations.

Step 4: Establish Final Conclusion About Convergence

Since the absolute value of each term \(|a_n| = \frac{1}{n}\) tends to zero as \(n\) approaches infinity, the sequence itself is convergent. The limit of the sequence is thus zero.

Key Concepts

Alternating Sequences

An alternating sequence is a type of sequence where the signs of its terms change from positive to negative or vice versa as you progress through the sequence. In the given sequence \[a_n = \frac{(-1)^n}{n},\]the \((-1)^n\) factor is responsible for this behavior. This alternating component is due to the exponential factor \((-1)^n\), which is positive when \(n\) is even and negative when \(n\) is odd.
Thus, the sequence fluctuates between positive and negative. This fluctuation does not affect whether the sequence converges or not. Alternating sequences are quite interesting because they can approach a limit even if they continuously change signs. The central question is whether these fluctuations diminish, allowing the sequence to converge.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index \(n\) becomes very large. For convergence, it is not sufficient just to have a single value that the sequence approaches; this value must be unique.
In the sequence \(a_n = \frac{(-1)^n}{n},\) as \(n\) tends to infinity, \(\frac{1}{n}\) approaches zero. This is because as the denominator \(n\) increases, the value of \(\frac{1}{n}\) becomes smaller and smaller, getting arbitrarily close to zero. Regardless of the sign changes introduced by \((-1)^n\), the magnitude \(|a_n|\) of each term continues to decrease towards zero. Thus, we find that the limit of this sequence is zero, establishing that the sequence is convergent.

Sequence Behavior Analysis

Analyzing the behavior of a sequence involves inspecting how its terms evolve as the index \(n\)\ increases. For the sequence \(a_n = \frac{(-1)^n}{n},\) it's noted that:

• When \(n\) is odd, \(a_n\) is negative, and the sequence's terms appear as \(-1/n.\)
• When \(n\) is even, \(a_n\) is positive, appearing as \(1/n.\)
• Both positive and negative terms diminish in absolute value because \(1/n\) decreases.

Despite the changing signs, the key aspect to note is that each term's absolute value approaches zero, ensuring that the overall sequence converges to zero. Understanding this sequence's behavior helps in predicting its long-term behavior and is integral for identifying convergence trends. Such insights are crucial for mathematical analysis and problem-solving, providing a clearer view of the sequence's limiting behavior.