Problem 54 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\ln n}{\ln 2 n} $$

Step-by-Step Solution

Answer

The sequence converges to 1.

Step 1: Rewrite the Sequence

We start with the sequence \( a_{n} = \frac{\ln n}{\ln 2n} \). To simplify, recognize that the denominator can be rewritten using a logarithmic identity: \( \ln 2n = \ln 2 + \ln n \). This gives us the sequence: \[ a_{n} = \frac{\ln n}{\ln 2 + \ln n} \].

Step 2: Simplify the Expression

Divide the numerator and the denominator by \( \ln n \) to simplify the sequence: \[ a_{n} = \frac{1}{\frac{\ln 2}{\ln n} + 1} \].

Step 3: Analyze the Limit as n Approaches Infinity

As \( n \to \infty \), \( \ln n \to \infty \) which makes \( \frac{\ln 2}{\ln n} \to 0 \). Therefore, the expression simplifies to \[ a_{n} \to \frac{1}{0 + 1} = 1 \].

Step 4: Conclude the Convergence

The sequence \( \left\{ a_{n} \right\} = \frac{\ln n}{\ln 2n} \) converges to 1 as \( n \to \infty \). Hence, the sequence is convergent.

Key Concepts

Limits of Sequences

Understanding the limits of sequences is fundamental when studying the convergence or divergence of a sequence. A sequence is a set of numbers in a specific order. When we say a sequence converges, we mean that as you progress through the sequence, its terms approach a specific, finite number called the limit.
The limit essentially tells us the "end behavior" of the sequence, reflecting what happens as the terms extend indefinitely. Mathematically, if the sequence \( \{a_n\} \), approaching a number \( L \) as \( n \) goes to infinity, is written as:

• \( \lim_{{n \to \infty}} a_n = L \)

In our exercise with the sequence \( a_n = \frac{\ln n}{\ln 2n} \), we simplified this to get \( a_n = \frac{1}{\frac{\ln 2}{\ln n} + 1} \). As \( n \to \infty \), \( \ln n \) grows indefinitely, making \( \frac{\ln 2}{\ln n} \rightarrow 0\). Hence, \( \lim_{{n \to \infty}} a_n = \frac{1}{0+1} = 1 \), showing that this sequence converges to 1.
Understanding the behavior of sequences at infinity is crucial for solving problems related to convergence and divergence.

Logarithmic Identities

Logarithmic identities come in handy when simplifying terms involving logarithms. They help break down complex expressions into simpler forms, making it easier to find limits or solve equations. In this problem, we used the identity \( \ln ab = \ln a + \ln b \). This allowed us to rewrite \( \ln 2n \) as:

• \( \ln 2n = \ln 2 + \ln n \)

By substituting this into the sequence \( a_n = \frac{\ln n}{\ln 2n} \), we simplified it further to \( a_n = \frac{1}{\frac{\ln 2}{\ln n} + 1} \).
This greatly clarifies the sequence's infinite limit behavior and facilitates the analysis of convergence. Logarithmic identities are valuable tools. They simplify sequences, uncovering hidden patterns necessary for thorough mathematical analysis and problem-solving.

Infinite Limit Behavior

Exploring infinite limit behavior reveals how sequences behave as they extend towards infinity. This concept is crucial as it allows mathematicians to predict the "tail behavior" of sequences. The primary focus is understanding how the terms stabilize, if at all, over their indefinite progression.
For the sequence \( a_n = \frac{\ln n}{\ln 2n} \), we discovered that as \( n \to \infty \), \( \ln n \to \infty \) and \( \frac{\ln 2}{\ln n} \rightarrow 0 \). Consequently, the entire expression simplifies to \( \frac{1}{0 + 1} = 1 \). This behavior indicates convergence, which means the values will settle to 1 as \( n \) increases without bound.
Identifying such behavior is key for discerning convergence patterns, especially in sequences and series. It provides insight into whether the terms stabilize or oscillate, giving an understanding of the mathematical journey they embark on.