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Chapter 5: Problem 352

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}$$

Short Answer

Expert verified
The series diverges by the Limit Comparison Test with \( b_n = 1 \).

Step by step solution

01

Understand the Problem

We need to determine whether the infinite series \( \sum_{n=1}^{\infty} \left(1+\frac{1}{n^{2}}\right)^{n} \) converges or diverges. The series involves the term \( a_n = \left(1+\frac{1}{n^{2}}\right)^{n} \).
02

Simplify the General Term

To analyze convergence, simplify \( a_n \). Use the approximation for the exponent: \( \left(1+\frac{1}{n^{2}}\right)^{n} \approx e^{1/n} \) as \( n \to \infty \). This gives that \( a_n \approx e^{1/n} \to 1 \) as \( n \to \infty \).
03

Apply Limit Comparison Test

We attempt the Limit Comparison Test with \( b_n = 1 \). Calculate \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} e^{1/n} = e^0 = 1 \). Since this does not equal 0, the convergence of \( \sum a_n \) follows the behavior of \( \sum b_n \).
04

Determine the Convergence of Reference Series

The series \( \sum b_n = \sum 1 \) is a divergent series because it is a constant series where each term is 1. Therefore, the series \( \sum a_n \) is also divergent.

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Key Concepts

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

Convergence Tests
Convergence tests are essential tools for determining whether a series converges or diverges. They help us understand if the sum of an infinite sequence approaches a finite limit or grows without bound. Various tests are used based on the type of series being analyzed. Some popular convergence tests include:
  • Geometric Series Test: Applicable for series of the form \( a r^n \). The series converges if the absolute value of the common ratio, \(|r|\), is less than 1.
  • p-Series Test: Used for series of the form \( \frac{1}{n^p} \). The series converges if \( p > 1 \).
  • Comparison Test: Compares the given series with another series whose convergence is already known.
  • Integral Test: Relates the convergence of a series to a corresponding improper integral.
No single test works for all series, so choosing the right test depends on the characteristics of the series. In the context of the given problem, where the series involves exponential terms, the Limit Comparison Test was a suitable choice.
Limit Comparison Test
The Limit Comparison Test is a method used to determine the convergence of a series by comparing it to another series with known behavior. If a series \( \sum a_n \) is compared with a series \( \sum b_n \), the test stipulates that:
  • Calculate the limit: \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
    • If the limit is a positive, finite number, then both series converge or both diverge.
    • If the limit is zero and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
    • If the limit is infinite and \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.
In the original exercise, the sequence \( a_n = \left(1+\frac{1}{n^2}\right)^n \) was compared to \( b_n = 1 \). The calculated limit was 1, leading to the conclusion that both series had the same convergence behavior. Since \( \sum b_n \) diverges, it follows that \( \sum a_n \) also diverges. This illustrates how the test offers a reliable way to infer the convergence properties of complex series.
Exponential Approximation
Exponential approximation is an essential technique in understanding the behavior of complex functions, especially those involving limits. For functions like \( \left(1+\frac{1}{n^2}\right)^n \), which appear in infinite series, approximations help simplify the terms for analysis.
  • The expression \( \left(1+\frac{1}{n^2}\right)^n \) can be approximated using the formula \( e^x \approx 1 + x\) for small values of \( x \), leading to \( \left(1+\frac{1}{n^2}\right)^n \approx e^{1/n} \).
  • As \( n \to \infty \), \( e^{1/n} \to 1 \), indicating that the terms of the series approach a constant value.
Such approximations simplify the examination of series by highlighting dominant trends and behaviors, which helps in selecting appropriate convergence tests. Approximations are also useful in error analysis and numerical solutions, making them crucial in both theoretical and practical applications of calculus.

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Most popular questions from this chapter

State whether each of the following series converges absolutely, conditionally, or not at all. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{1+n^{4}}$$

In each of the following problems, use the estimate \(\left|R_{N}\right| \leq b_{N+1}\) to find a value of \(N\) that guarantees that the sum of the first \(N\) terms of the alternating series \(\sum_{n=1}^{\infty}(-1)^{n+1} b_{n}\) differs from the infinite sum by at most the given error. Calculate the partial sum \(S_{N}\) for this \(N\). $$\text { [T] } b_{n}=1 / n^{2}, \text { error }<10^{-6}$$

Let \(a_{n}=2^{-[n / 2]}\) where \([x]\) is the greatest integer less than or equal to \(x\). Determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges and justify your answer. The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges.

Use the root and limit comparison tests to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges. $$a_{n}=1 / x_{n}^{n} \quad\( where \)\quad x_{n+1}=\frac{1}{2} x_{n}+\frac{1}{x_{n}}, \quad x_{1}=1$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{n}=\frac{(\ln (1+\ln n))^{n}}{(\ln n)^{n}}$$
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