Problem 64 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\ln n-\ln (n+1) $$

Step-by-Step Solution

Answer

The sequence converges, and the limit is 0.

Step 1: Simplify the Sequence Term

The given sequence is \( a_n = \ln n - \ln (n+1) \). We can simplify this expression using the logarithmic identity \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). Applying this identity, we get:\[a_n = \ln \left( \frac{n}{n+1} \right)\]

Step 2: Analyze the Expression \( \frac{n}{n+1} \) as \( n \to \infty \)

Consider the expression \( \frac{n}{n+1} \). As \( n \to \infty \), the denominator \( n+1 \) becomes very close to \( n \). This means:\[\frac{n}{n+1} \to 1\]

Step 3: Find the Limit of \( a_n \) as \( n \to \infty \)

Since \( \ln(1) = 0 \), we need to determine the behavior of \( \ln \left( \frac{n}{n+1} \right) \) as \( n \to \infty \). As \( \frac{n}{n+1} \to 1 \), the logarithm of a value approaching 1 is approaching 0:\[\lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} \ln \left( \frac{n}{n+1} \right) = \ln(1) = 0\]

Step 4: Conclusion on Convergence

Since the limit of \( a_n \) as \( n \to \infty \) is 0, the sequence \( \{a_n\} \) converges to 0.

Key Concepts

Logarithmic Identity

Logarithmic identities are mathematical tools that help simplify expressions involving logarithms. One crucial identity to remember is the difference of two logarithms:

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

This identity is useful when you encounter a difference such as \( \ln n - \ln (n+1) \). By applying it, we rewrite the expression as a single logarithm. This transformation simplifies complex expressions and provides a clearer path towards finding the limit of a sequence. Converting differences into fractions within a logarithm often reveals the behavior of a sequence much more clearly. This identity is extensively used in calculus and algebra to solve problems involving growth rates, exponential functions, and sequences.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus. It describes where the terms of a sequence "head" as the index becomes very large. Suppose you have a sequence defined by \( a_n \). You would want to know what happens to \( a_n \) as \( n \) becomes infinitely large.

• The formal notation is \( \lim_{{n \to \infty}} a_n = L \), where \( L \) is the value the sequence approaches.

In the given sequence \( a_n = \ln \left( \frac{n}{n+1} \right) \), we observe how the fraction \( \frac{n}{n+1} \) behaves. As \( n \to \infty \), \( \frac{n}{n+1} \to 1 \).
Applying this in the logarithm, \( \ln \left( \frac{n}{n+1} \right) \to \ln(1) = 0 \). Hence, the sequence converges to 0. Understanding sequence limits helps in analyzing series and determining convergence or divergence behavior efficiently.

Infinite Series Convergence

Infinite series convergence is about determining whether an infinite sum of terms ends up being a finite number. Convergence is crucial in calculus and analysis whenever we're summing an infinite list of numbers.

• A series converges if its sequence of partial sums approaches a limit.
• Divergence means the sequence of sums doesn't settle at any limit.

In our scenario, we aren't dealing directly with an infinite series, but understanding convergence principles of sequences like \( a_n = \ln \left( \frac{n}{n+1} \right) \) is essential. A convergent sequence can lead to a convergent series, where each term is part of an infinite sum. Knowing the limit of a sequence, as we know from \( a_n \to 0 \), helps inform us about the "fate" of potential series formed from its terms. Proper grasp of convergence allows better comprehension of how such series behave, especially when predicting values and solving calculus problems.