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Chapter 10: Problem 15

Use the Root Test to determine if each series converges absolutely or diverges. $$\begin{aligned} &\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{1}{n}\right)^{n^{2}}\\\ &\text { (Hint: } \lim _{x \rightarrow 0}(1+x / n)^{n}=e^{x} \end{aligned}$$

Short Answer

Expert verified
The series converges absolutely.

Step by step solution

01

Identify the Root Test

The Root Test is used to determine the convergence of a series. For a series \( \sum a_n \), we calculate \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If the limit \( L < 1 \), the series converges absolutely. If \( L > 1 \), the series diverges. If \( L = 1 \), the test is inconclusive.
02

Define \(a_n\) for the Series

In this series, \( a_n = (-1)^n \left(1 - \frac{1}{n}\right)^{n^2} \). We focus on \( |a_n| = \left(1 - \frac{1}{n}\right)^{n^2} \) since the Root Test uses absolute values, neglecting the alternating sign.
03

Calculate the nth Root of \( |a_n| \)

To apply the Root Test, find \( \sqrt[n]{|a_n|} \):\[ \sqrt[n]{\left(1 - \frac{1}{n}\right)^{n^2}} = \left(1 - \frac{1}{n}\right)^{n} \] by simplifying the expression using properties of roots and exponents.
04

Compute the Limit

Calculate the limit \( \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n} \). By the hint and a known limit, this expression converges to \( e^{-1} \) as \( n \to \infty \).
05

Conclusion Using the Root Test

Since \( e^{-1} \approx 0.3679 \) is less than 1, the series \( \sum_{n=1}^{\infty} (-1)^n \left(1 - \frac{1}{n}\right)^{n^2} \) converges absolutely by the Root Test.

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Key Concepts

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

Series Convergence
Series convergence is a fundamental concept in calculus and analysis. It helps us understand whether the sum of an infinite sequence of terms results in a finite value. To determine convergence, different tests are applied depending on the series' structure. One popular test is the Root Test. This test checks how fast the terms of the series shrink as the series progresses. With our given series, convergence indicates that the infinite sum behaves nicely and doesn't blow up to infinity. In other words, if a series converges, you can think of adding more and more terms without the sum going to infinity.
Absolute Convergence
Absolute convergence is a stronger form of convergence. When a series converges absolutely, it means the series converges even when you consider the absolute values of its terms. A series is said to converge absolutely if the series of its absolute values is convergent. This implies:
  • If a series converges absolutely, it will also converge in the regular sense (conditional convergence), but not vice versa.
  • Many convergence tests, like the Root Test, work with absolute values. Absolute convergence is preferred because it guarantees stability; rearranging terms won't affect the sum.
In this exercise, because the Root Test evaluates the absolute values, once we determine absolute convergence, the series is robust to various operations.
Limit Computations
Computing limits forms the backbone of many convergence tests, including the Root Test. Here, the limits help simplify expressions and determine if they are finite and what value they converge to. Typically, computing limits involves recognizing familiar patterns or applying known limits. In our exercise:
  • We derive an expression to test by finding the n-th root of absolute terms.
  • Use a known limit: \( \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n} = e^{-1} \), which plays a crucial role in determining convergence.
Effectively handling limit computations allows understanding whether the series ultimately converges or diverges.

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Most popular questions from this chapter

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) \(1 /(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)

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 \(L ?\) \(a_{n}=n \sin \frac{1}{n}\)

Use a geometric series to represent each of the given functions as a power series about \(x=0,\) and find their intervals of convergence. a. \(f(x)=\frac{5}{3-x}\) b. \(\quad g(x)=\frac{3}{x-2}\)

a. Assuming that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant, show that $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$$ if \(c\) is any positive constant. b. Prove that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant. (Hint: If \(\epsilon=0.001\) and \(c=0.04,\) how large should \(N\) be to ensure that \(\left.\left|1 / n^{c}-0\right|<\epsilon \text { if } n>N ?\right)\)

If \(\Sigma a_{n}\) converges and \(\Sigma b_{n}\) diverges, can anything be said about their term-by-term sum \(\Sigma\left(a_{n}+b_{n}\right) ?\) Give reasons for your answer.
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