Problem 24 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Identifying the Sequence

The sequence given is \(a_n = \ln(n+1) - \ln n\). This is a difference of logarithms which can be simplified using logarithmic identities.

Step 2: Simplify using Logarithmic Identities

Recall the logarithmic identity: \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\). Using this, \(a_n = \ln\left(\frac{n+1}{n}\right)\).

Step 3: Rewrite the Fraction

The fraction \(\frac{n+1}{n}\) can be rewritten as \(1 + \frac{1}{n}\). Thus, \(a_n = \ln\left(1 + \frac{1}{n}\right)\).

Step 4: Apply the Limit Process

As \(n\) approaches infinity, evaluate the limit \(\lim_{{n \to \infty}} \ln\left(1 + \frac{1}{n}\right)\). We know that \(1 + \frac{1}{n}\) approaches 1.

Step 5: Finding the Limit

As \(n\) approaches infinity, \(\ln(1 + \frac{1}{n})\) approaches \(\ln(1)\), which is 0. You can also use the property that \(\lim_{{x \to 0}} \ln(1+x) = x\) when \(x\) is small, to confirm this.

Key Concepts

Logarithmic Identities

In mathematics, logarithmic identities are useful tools that help simplify complex expressions involving logarithms. One key identity is

• \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \).

This identity allows us to combine two separate logarithms into one by converting a difference into the logarithm of a quotient.
In the given sequence, \( a_n = \ln(n+1) - \ln n \), this identity simplifies the expression to

• \( a_n = \ln\left(\frac{n+1}{n}\right) \).

This simplification is crucial because it makes further analysis easier, especially when dealing with sequences and limits. Understanding and applying these logarithmic identities are essential in calculus and algebra to transition from complex to more manageable forms of mathematical expressions.

Limit Process

The limit process deals with understanding the behavior of sequences or functions as variables approach a certain point, often infinity. For sequences, we examine what value, if any, the terms approach as the sequence progresses endlessly.
In the exercise, we used the sequence \( a_n = \ln\left(1 + \frac{1}{n}\right) \). As \( n \to \infty \), the expression \( \frac{1}{n} \) becomes very small, approaching zero.

• Thus, \( 1 + \frac{1}{n} \) approaches \( 1 \).
• And \( \ln(1) \) is \( 0 \).

Hence, we conclude that the sequence converges. The limit of the sequence's terms as \( n \to \infty \) is \(0\).
This result can also be verified using the logarithm approximation: \( \ln(1+x) \approx x \) when \( x \) is small. This reveals that as \( n \to \infty \), the expression \( \ln\left(1 + \frac{1}{n}\right) \) approaches \( \frac{1}{n} \), which again goes to \( 0 \). This process is fundamental in calculus for analyzing how functions behave at extreme values.

Infinite Sequences

Infinite sequences are ordered lists of numbers that continue indefinitely. Understanding the behavior of these sequences is important, particularly in determining whether they converge (approach a specific value) or diverge (do not approach any single value).In our specific case, the sequence \( a_n = \ln(n+1) - \ln n \) simplifies to \( a_n = \ln\left(1 + \frac{1}{n}\right) \).
As \( n \) increases, investigating whether the terms in the sequence converge or diverge is essential. We applied the concept of limits to see that the sequence converges to \( 0 \).

• Recognizing convergence is key in calculus, as it helps define integral concepts such as the limit and continuity of functions.
• Understanding that infinity is not a number but a concept helps students avoid common pitfalls when dealing with infinite sequences.

Being able to predict whether a sequence converges or diverges allows mathematicians and students to navigate and solve problems involving infinite series and sequences with greater clarity.