Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

An explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{\ln n}{\sqrt{n}} $$

Step-by-Step Solution

Answer

The sequence converges, and the limit is 0.

Step 1: Write the First Term

Substitute \(n = 1\) into \(a_n = \frac{\ln n}{\sqrt{n}}\) to get the first term.\[a_1 = \frac{\ln 1}{\sqrt{1}} = \frac{0}{1} = 0\]

Step 2: Write the Second Term

Substitute \(n = 2\) into \(a_n = \frac{\ln n}{\sqrt{n}}\) to find the second term.\[a_2 = \frac{\ln 2}{\sqrt{2}} \approx \frac{0.693}{1.414} \approx 0.490 \]

Step 3: Write the Third Term

Substitute \(n = 3\) into \(a_n = \frac{\ln n}{\sqrt{n}}\) to get the third term.\[a_3 = \frac{\ln 3}{\sqrt{3}} \approx \frac{1.099}{1.732} \approx 0.635 \]

Step 4: Write the Fourth Term

Substitute \(n = 4\) into \(a_n = \frac{\ln n}{\sqrt{n}}\) for the fourth term.\[a_4 = \frac{\ln 4}{\sqrt{4}} = \frac{1.386}{2} \approx 0.693 \]

Step 5: Write the Fifth Term

Substitute \(n = 5\) into \(a_n = \frac{\ln n}{\sqrt{n}}\) for the fifth term.\[a_5 = \frac{\ln 5}{\sqrt{5}} \approx \frac{1.609}{2.236} \approx 0.720 \]

Step 6: Analyze Convergence or Divergence

Consider the behavior of \(a_n = \frac{\ln n}{\sqrt{n}}\) as \(n\) approaches infinity. Observe that \(\ln n\) grows slower than any polynomial, but with the denominator increasing faster, the quotient will approach zero.As \(n \rightarrow \infty\), \(\frac{\ln n}{\sqrt{n}} \rightarrow 0\). The sequence converges to 0.

Step 7: Conclusion on the Limit

The sequence does converge, and the limit is given by:\[ \lim_{n \to \infty} a_n = 0 \]

Key Concepts

Explicit Formula

An explicit formula is a mathematical expression that allows us to compute the terms of a sequence directly, for any position \(n\), without needing to reference previous terms. In the given exercise, the explicit formula for the sequence \(a_n\) is \(a_n = \frac{\ln n}{\sqrt{n}}\). This formula lets us calculate the value of each term by simply plugging in the value of \(n\). This kind of formula is especially useful because it provides a straightforward computational approach as opposed to relying on recursion or iteration. With explicit formulas, we can quickly obtain terms in the sequence, which is crucial for further analysis, like determining convergence or finding limits. Explicit formulas are key in activities such as generating terms or analyzing the pattern of the sequence with ease.

Sequence Terms

The sequence terms are the actual values you obtain from a sequence's explicit formula when substituting various values of \(n\). For the exercise, the task was to determine the first five terms of the sequence \(a_n = \frac{\ln n}{\sqrt{n}}\). By substituting \(n = 1\) through \(n = 5\), we calculated the terms as follows:

• \(a_1 = 0\)
• \(a_2 \approx 0.490\)
• \(a_3 \approx 0.635\)
• \(a_4 \approx 0.693\)
• \(a_5 \approx 0.720\)

Each of these terms gives us a snapshot of the sequence at a particular index. As part of sequence analysis, examining these individual terms helps us understand the behavior and trend of the sequence. It also provides the initial groundwork for understanding broader characteristics such as convergence and limits.

Limit of a Sequence

The limit of a sequence describes the value that the terms of the sequence approach as \(n\) becomes very large (approaches infinity). In this problem, we need to explore the behavior of \(a_n = \frac{\ln n}{\sqrt{n}}\) as \(n\) grows larger. Although the natural logarithm, \(\ln n\), increases with \(n\), it does so at a slower rate compared to the increase of \(\sqrt{n}\). This results in \(\ln n\) being outpaced by \(\sqrt{n}\), causing the ratio \(\frac{\ln n}{\sqrt{n}}\) to approach zero. Thus, the sequence converges, and the limit is:
\[\lim_{n \to \infty} a_n = 0\]
Understanding this concept is crucial because it informs us about the behavior of infinite sequences and whether they settle at a specific value as the number of terms increases. The existence of a limit often implies desirable properties in analysis, like convergence, which simplifies further mathematical applications.

Natural Logarithm in Sequences

The natural logarithm, denoted \(\ln n\), is a critical mathematical function in many areas, including sequence analysis. It describes the growth rate of sequences in logarithmic terms. For large \(n\), \(\ln n\) increases but at a slower pace compared to polynomial functions or even some roots like \(\sqrt{n}\).
In the sequence \(a_n = \frac{\ln n}{\sqrt{n}}\), it represents the rate of increase for the numerator, which affects the overall behavior of the sequence. The influence of \(\ln n\) here is crucial because, although it's growing, its growth is not sufficient to prevent the entire expression \(\frac{\ln n}{\sqrt{n}}\) from diminishing to zero as \(n\) becomes very large.

• \(\ln n\) grows slower than polynomial functions.
• It manages the balance between growth and decay in complex sequences.

Comprehending the behavior of \(\ln n\) is essential for accurately assessing whether sequences containing logarithmic elements converge or diverge. It shows how sequences comprising logarithmic terms can still converge based on the relative rates of change in numerator and denominator components.