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Chapter 10: Problem 68

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

Short Answer

Expert verified
The sequence converges, and the limit is 1.

Step by step solution

01

Rewrite the Sequence

Given the sequence \(a_n = \ln \left(1 + \frac{1}{n}\right)^n\), we can rewrite it using the power property of logarithms: \(a_n = n \ln \left(1 + \frac{1}{n}\right)\).
02

Apply the Limit Definition

To find the limit of the sequence as \(n\) approaches infinity, consider \(L = \lim_{n \to \infty} n \ln \left(1 + \frac{1}{n}\right)\). This resembles the definition of \(e\).
03

Use Exponential Limit Property

Recall the limit definition of the exponential function: \(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e\). Therefore, we know that the limit of \(n \ln \left(1 + \frac{1}{n} \right) = \ln \left(\left(1 + \frac{1}{n}\right)^n\right)\) approaches \(\ln(e) = 1\).
04

Convergence and Limit Verification

Since the limit \(L = 1\) exists, the sequence \(a_n\) is convergent. Therefore, the sequence converges to \(1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Sequence
The concept of a sequence limit is crucial in understanding convergence. When we analyze limits, we're looking to see what value a sequence approaches as the index (often represented as \(n\)) becomes infinitely large. To say a sequence \(\{a_n\}\) converges, means that there exists a number \(L\) such that the terms \(a_n\) get arbitrarily close to \(L\) as \(n\) increases. Mathematically, this is written as \(\lim_{n \to \infty} a_n = L\). If such a limit does not exist, the sequence is said to diverge.
  • Convergent sequences have a finite limit.
  • Divergent sequences do not approach a specific value.
Once a sequence is confirmed to converge, finding the limit \(L\) gives us valuable information about its long-term behavior.
Logarithmic Limit
Logarithmic limits involve sequences where terms include logarithmic expressions. For example, consider \(a_n = n \ln\left(1 + \frac{1}{n}\right)\). Using properties of logarithms can simplify finding limits:
  • The Chain property: \(\ln(a^b) = b \cdot \ln(a)\).
  • The Limit property: \(\lim_{n \to \infty} \ln(f(n)) = \ln(L)\) if \(\lim_{n \to \infty} f(n) = L\).
In this exercise, rewriting the sequence helps us recognize the logarithmic limit as part of a well-known approximation to \(e\), Euler's number. The sequence eventually is simplified to follow the pattern of logarithms applied to an expression known to converge.
Exponential Limit
The exponential limit is a fundamental concept often related to Euler's number \(e\). A famous limit that defines \(e\) is \(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e\). Recognizing sequences that fit this template can simplify solving them as shown in the original exercise.
In explaining the sequence \(a_n\), we found that \(n \ln\left(1 + \frac{1}{n}\right)\) simplifies to the expression approaching \(\ln(e) = 1\). The exponential limit highlights how combinations of logarithmic and power expressions can converge to key mathematical constants.
  • Knowing basic limits, such as those defining \(e\), streamlines solving complex limits.
  • Patterns in sequences often relate to well-studied limits.
Sequence Divergence and Convergence
A sequence either converges to a limit or diverges, and recognizing this can determine the next steps in analysis. In looking at \(a_n = \ln \left(1 + \frac{1}{n}\right)^n\), through simplification and application of known limits, we deduced convergence.
Convergence means that as \(n\) becomes very large, \(a_n\) approaches a specific number, here, 1. Divergence, on the other hand, implies the sequence doesn't settle to a limit but may increase indefinitely or oscillate.
  • Understanding convergence lets us make predictions about sequence behavior.
  • Divergence requires further examination to understand sequence properties.
In practice, determining whether a sequence converges is often the primary question, followed by finding the limit itself if convergence is confirmed.

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Most popular questions from this chapter

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=n \sin \frac{1}{n}\)

A sequence of rational numbers is described as follows: \(\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots\) Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let \(x_{n}\) and \(y_{n}\) be, respectively, the numerator and the denominator of the \(n\) th fraction \(r_{n}=x_{n} / y_{n}\) a. Verify that \(x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1\) and, more generally, that if \(a^{2}-2 b^{2}=-1\) or \(+1,\) then \((a+2 b)^{2}-2(a+b)^{2}=+1 \quad\) or \(\quad-1\) respectively. b. The fractions \(r_{n}=x_{n} / y_{n}\) approach a limit as \(n\) increases. What is that limit? (Hint: Use part (a) to show that \(\left.r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2} \text { and that } y_{n} \text { is not less than } n .\right)\)

Assume that each sequence converges and find its limit. \(a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2}\)

Prove that \(\lim _{n \rightarrow \infty} x^{1 / n}=1,(x>0)\).

Use the definition of convergence to prove the given limit. \(\lim _{n \rightarrow \infty} \frac{\sin n}{n}=0\)
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