Problem 18 - Essential Calculus Early Transcendentals

Question

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{(-1)^{n+1} n}{n+\sqrt{n}}$$

Step-by-Step Solution

Answer

The sequence diverges as it oscillates between 1 and -1.

Step 1: Analyzing the Sequence Behavior

Given the sequence \( a_n = \frac{(-1)^{n+1} n}{n + \sqrt{n}} \), we first note that the numerator and denominator increase as \( n \) becomes large. However, the numerator alternates between \( n \) and \(-n\), suggesting that the sequence alternates in sign.

Step 2: Simplifying the Expression

For large values of \( n \), the term \( \sqrt{n} \) becomes small compared to \( n \). Hence, the denominator \( n + \sqrt{n} \approx n \). Simplifying the sequence yields \( a_n \approx (-1)^{n+1} \frac{n}{n} = (-1)^{n+1}. \)

Step 3: Determining Sequence Convergence or Divergence

Since \( (-1)^{n+1} \) does not approach a single value as \( n \) increases, the sequence \( a_n = (-1)^{n+1} \) does not converge. The values oscillate between \( 1 \) and \(-1\), indicating divergence.

Key Concepts

Understanding Oscillating Sequences

An oscillating sequence is one that does not settle down to a particular value but instead keeps switching, often in a regular pattern. This behavior is evident in our sequence, \(a_n = \frac{(-1)^{n+1} n}{n + \sqrt{n}}\), where the term \((-1)^{n+1}\) causes oscillation. This portion of the sequence flips the sign of the term with each step in \(n\), leading the sequence value to fluctuate between positive and negative.In simple terms, an oscillating sequence

• changes direction regularly, either increasing or decreasing.
• does not converge to a single limit, meaning it does not stabilize at a single number as \(n\) grows larger.

This sequence is a perfect example because, as explained in the exercise, the numerator alternates between \(n\) and \(-n\), driven by the \((-1)^{n+1}\) term. As a result, the values switch between 1 and -1, further demonstrating oscillation.

Dissecting Alternating Series

An alternating series is a type of series where the signs of the terms alternate, typically represented by a factor of \((-1)^n\) or \((-1)^{n+1}\). In our given sequence, we see the factor \((-1)^{n+1}\), producing an alteration between positive and negative terms.Here's how alternating behavior affects the sequence:

• The factor \((-1)^{n+1}\) changes the sign of each term, leading to the alternating pattern.
• This pattern impacts the series by giving rise to a characteristic 'zig-zag' graph if plotted, swinging between positive and negative territory.
• The alternating nature can prevent the series from converging, as consistent oscillation impedes settling towards a specific limit.

Understanding alternating series helps us see why such sequences often fail to converge—they're pulled in different directions with each step, so they never stabilize at a single value.

Analyzing Limits of Sequences

The concept of a sequence's limit is fundamental in understanding convergence. A sequence converges if its terms settle down to a single value as \(n\) approaches infinity. For the sequence \(a_n = \frac{(-1)^{n+1} n}{n + \sqrt{n}}\), we determine its convergence by evaluating its limit as \(n\) moves toward infinity.Let's break it down:

• As \(n\) grows large, the \(\sqrt{n}\) in the denominator becomes negligible compared to \(n\). This simplifies the expression to \(a_n \approx (-1)^{n+1}\).
• The expression \((-1)^{n+1}\) itself does not settle down to a specific number. Rather, it oscillates indefinitely between \(1\) and \(-1\).
• This oscillation indicates that \(a_n\) lacks a limit, and thus, the sequence diverges.

In summary, the sequence does not have a limit because its alternating nature doesn't let it stabilize, ultimately showing that the sequence diverges.