Problem 28 - Precalculus Mathematics 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 converges to 0.

Step 1: Understand the Sequence

The given sequence is \( a_n = \frac{(-1)^n}{n} \). This sequence involves the term \((-1)^n\), which causes the sign of each term to alternate based on the value of \(n\). The denominator \(n\) indicates that as \(n\) increases, the size of each term should decrease.

Step 2: Recognize the Behavior of Terms

Note that \((-1)^n\) changes the sign of the sequence's terms with each increase in \(n\): odd \(n\) gives negative values, and even \(n\) gives positive values. However, the numerator's absolute value is always 1, and the denominator grows with \(n\). This means the absolute value of the terms is \( \frac{1}{n} \), which approaches zero as \(n\) goes to infinity.

Step 3: Determine Convergence or Divergence

A sequence is convergent if there exists a limit \(L\) such that \( \lim_{{n \to \infty}} a_n = L \). Evaluate \( \lim_{{n \to \infty}} (-1)^n/n \). Since the absolute value of this term, \(1/n\), approaches zero, \( a_n \) approaches zero despite the alternating sign. Thus, \( \lim_{{n \to \infty}} a_n = 0 \), indicating that the sequence converges.

Step 4: Conclude with the Limit

The alternating sign does not affect convergence to the limit point zero, as the magnitude of each term is evenly going toward zero. Therefore, the sequence \( a_n = \frac{(-1)^n}{n} \) is convergent with the limit \(0\).

Key Concepts

Alternating Sequence

An alternating sequence is a type of sequence where the terms change signs in a regular pattern as the sequence progresses. This usually involves terms that switch between positive and negative values. In mathematical notation, an alternating sequence can be represented as \((-1)^n a_n\), where each term flips its sign based on whether \(n\) is odd or even.

In the given exercise, the alternating sequence is represented by the formula \(a_n = \frac{(-1)^n}{n}\). This means:

• For odd values of \(n\) (e.g., 1, 3, 5), \((-1)^n = -1\), resulting in negative terms.
• For even values of \(n\) (e.g., 2, 4, 6), \((-1)^n = 1\), resulting in positive terms.

Alternate sign patterns are essential to recognize as they affect how fast a sequence converges or transitions between values. Although the signs change, this doesn't automatically imply divergence. Instead, we examine whether the sequence's magnitude is getting closer to a specific point as \(n\) tends to infinity.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index \(n\) goes to infinity. A sequence has a limit \(L\) if, after some point in the sequence, all of its terms remain arbitrarily close to \(L\). In mathematical terms, this is written as:\[\lim_{{n \to \infty}} a_n = L\]Where \(a_n\) is the nth term of the sequence.

In the example of \( a_n = \frac{(-1)^n}{n} \), we observe the magnitude of \(\frac{1}{n}\), which decreases to zero as \(n\) increases. Despite the alternating sign, this effect doesn't hinder the sequence's convergence to its limit. Consequently, we establish that the limit:\[\lim_{{n \to \infty}} \frac{(-1)^n}{n} = 0\]Thus, even with the fluctuations in sign, the proximity of terms to zero characterizes the behavior of the sequence as it moves towards its limit.

Sequence Convergence

Sequence convergence is when a sequence approaches a particular value, termed the limit, as the number of terms increases indefinitely. For a sequence to converge, its terms must get increasingly close to a single number. This converging behavior can be verified through the definition of limit, which involves checking if the distance between sequence terms and the limit becomes arbitrary small.

In the case of the sequence \( a_n = \frac{(-1)^n}{n} \):

• The absolute value, \( \frac{1}{n} \), diminishes as \(n\) grows, ignoring the sign change.
• Thus, each term of the sequence gets closer to zero, the specified limit.

It’s important to note that convergence is not disrupted by alternating signs, provided that the absolute value of the terms tends towards zero. The convergence here to the limit zero is confirmed by the steady reduction to the distance of terms from this value, illustrating a typical case of convergence in an alternating sequence.