Problem 27 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=2+(0.1)^{n}$$

Step-by-Step Solution

Answer

The sequence converges to 2.

Step 1: Understanding the Sequence

We are given the sequence \(a_n = 2 + (0.1)^n\). Here, \(n\) is a natural number, starting from 1, and \((0.1)^n\) is the expression raised to the \(n\)th power.

Step 2: Analyzing the Converging Behavior

To determine if the sequence converges or diverges, observe the behavior of \((0.1)^n\) as \(n\) approaches infinity. Since \(0.1\) is a number between 0 and 1, \((0.1)^n\) approaches 0 as \(n\) becomes very large.

Step 3: Determining the Limit

Since \((0.1)^n\) approaches 0 as \(n\) approaches infinity, the sequence \(a_n = 2 + (0.1)^n\) will approach \(2 + 0 = 2\). Thus, the sequence converges to 2.

Key Concepts

Limit of a Sequence

The concept of a "limit of a sequence" is foundational in understanding sequence convergence. When we talk about the limit of a sequence, we're asking what value the terms of the sequence approach as the sequence progresses to infinity. More formally, a sequence \( \{a_n\} \) has a limit \( L \) if, for every positive number \( \epsilon \), no matter how small, there exists a natural number \( N \) such that for all \( n \geq N \), the terms of the sequence are within \( \epsilon \) of \( L \).

• Visualize a sequence as a series of steps leading towards a specific point on a number line.
• The limit is the value that these steps are trying to reach.
• If the sequence keeps getting closer and closer to a number without ever "wobbling" off course, it is converging towards that number, which is the limit.

In the sequence \( a_n = 2 + (0.1)^n \), as \( n \) becomes very large, the term \((0.1)^n\) becomes negligible, converging to 0. Thus, the sequence itself approaches 2 as its limit.

Power Convergence

Power convergence refers to the behavior of terms raised to the power of an index, like \( (0.1)^n \) in our example sequence. Consider any base between 0 and 1, such as 0.1. Raising it to an increasing power (like \( n \)) results in progressively smaller values.

• This is because multiplying small fractions results in even smaller numbers.
• The more times you multiply 0.1 by itself, the closer the result gets to zero.

For instance, \( (0.1)^1 = 0.1 \), \( (0.1)^2 = 0.01 \), and \( (0.1)^3 = 0.001 \), continuing this pattern toward zero as \( n \) increases. This property of powers of numbers less than one is a key aspect of power convergence, making it crucial in determining the convergence of sequences like \( a_n = 2 + (0.1)^n \), which ultimately stabilizes to a limit, simplifying to \( a_n = 2 \).

Convergent Sequence Analysis

Analysis of sequence convergence involves understanding how the terms behave as the index progresses. A sequence is deemed convergent if it settles into a single, finite limit as the index (usually \( n \)) approaches infinity.

• This implies stability and predictability in the sequence's behavior.
• To analyze a sequence, consider the impact of each component term's behavior as \( n \) increases.
• Watch how rapidly terms approach a specific value, and if they stop deviating too much after a while, convergence is occurring.

In the case of \( a_n = 2 + (0.1)^n \), the analysis revealed that as \( (0.1)^n \) shrinks towards zero, the sequence reduces basically to the constant term 2, clearly indicating convergence. Thus, the convergent sequence analysis provides not just the limit, but the rationale for convergence, helping us fully comprehend the sequence's behavior.