Problem 28 - Precalculus: Mathematics for Calculus

Question

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{(-1)^{n}}{n}$$

Step-by-Step Solution

Answer

The sequence converges to 0 as \( n \rightarrow \infty \).

Step 1: Understand the Sequence Behavior

Let's first look at the sequence given: \( a_n = \frac{(-1)^n}{n} \). The presence of \((-1)^n\) implies that the sequence alternates in sign. This means for even \(n\), \(a_n\) is positive and for odd \(n\), \(a_n\) is negative.

Step 2: Analyze Absolute Values

Consider the absolute value of the sequence: \( |a_n| = \frac{1}{n} \). As \(n\) approaches infinity, \(\frac{1}{n} \) approaches 0. This indicates that the magnitude of terms becomes smaller and trends towards zero.

Step 3: Investigate Behavior at Infinity

Although the sequence alternates between positive and negative values, we see that each term \( a_n \) in absolute value becomes arbitrarily small as \( n \) increases, because \( |a_n| = \frac{1}{n} \rightarrow 0 \) as \( n \rightarrow \infty \).

Step 4: Convergence Conclusion

Despite the alternating signs, the overall trend of the sequence \( a_n = \frac{(-1)^n}{n} \) drives it towards zero as \( n \rightarrow \infty \). Therefore, the sequence converges to 0.

Key Concepts

Limits of Sequences

The concept of limits in sequences is crucial for understanding whether a sequence converges or diverges. To find the limit of a sequence means to determine what value, if any, the terms of the sequence approach as the index, often labeled as \(n\), goes towards infinity. If a sequence tends towards a specific number, we say it converges to that number, known as the limit.

When analyzing limits, consider the sequence given in the exercise: \(a_n = \frac{(-1)^n}{n}\). At first glance, it might seem challenging due to the alternating sign from \((-1)^n\). However, by focusing on its absolute values, \(|a_n| = \frac{1}{n}\), we observe a clear path. As \(n\) increases, \(\frac{1}{n}\) approaches \(0\). Thus, we conclude that the sequence \(a_n\) converges to \(0\) because the magnitude of its terms become negligible as \(n\) grows indefinitely.

Understanding limits helps to predict sequence behavior, one of the many stepping stones in calculus that encourages precise functions analysis and deeper insight into mathematical phenomena.

Alternating Sequences

Alternating sequences are marked by their terms' signs switching between positive and negative, and they are often denoted by the factor \((-1)^n\) or similar, which flips the sign for successive terms. This switch introduces an intriguing dimension to sequence analysis.

For the sequence \(a_n = \frac{(-1)^n}{n}\), we see this exact alternation:

• For even \(n\): the term \(a_n\) becomes positive.
• For odd \(n\): the term \(a_n\) becomes negative.

Analyzing this pattern, it’s evident how alternating sequences can lead to interesting behavior in the quest to find limits or convergence.
While analyzing alternating sequences, the changes in signs do not alter the magnitude, which is crucial in determining the limits and convergence. As seen with our sequence, despite the alternation, the \(|a_n| = \frac{1}{n}\) heads towards 0, making the alternating sign merely a feature rather than an obstacle to finding convergence.

Sequence Behavior

Understanding sequence behavior provides valuable insights into whether a series of numbers can seemingly stabilize at a particular value or not. One essential behavior is the convergence, where a sequence approaches an identifiable limit as \(n\) becomes very large.

The sequence \(a_n = \frac{(-1)^n}{n}\) illustrates fascinating behavior:

• It oscillates due to alternate signs but diminishes in magnitude over time.
• Each term magnitude decreases, following \(|a_n| = \frac{1}{n}\), which trends to zero as \(n\rightarrow \infty\).

This particular behavior hints toward convergence. Despite fluctuations between positive and negative, the sequence’s terms shrink towards zero, portraying a convergent sequence.
Sequences like these are incredibly useful in understanding mathematical concepts in calculus and other branches, where recognizing patterns and applying knowledge of limits and convergence guides to recognizing functions and series behavior, supporting accurate and meaningful conclusions in analysis.