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Chapter 9: Problem 37

Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)^{n}}{(2 n)^{n}}$$

Short Answer

Expert verified
The series converges absolutely.

Step by step solution

01

Identify the Nature of the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)^{n}}{(2n)^{n}} \). Since the series contains \((-1)^n\), it is an alternating series. To determine its behavior, we'll study both its absolute convergence and conditional convergence.
02

Check Absolute Convergence

For absolute convergence, consider \( \sum_{n=1}^{\infty} \left| \frac{(-1)^{n}(n+1)^{n}}{(2n)^{n}} \right| = \sum_{n=1}^{\infty} \frac{(n+1)^{n}}{(2n)^{n}} \). Simplify this expression:\[\frac{(n+1)^{n}}{(2n)^{n}} = \left(\frac{n+1}{2n}\right)^n = \left(\frac{1 + \frac{1}{n}}{2}\right)^n.\]Evaluate the limit of \( \left(\frac{1 + \frac{1}{n}}{2}\right)^n \) as \( n \to \infty \). This arises to:\[\lim_{n \to \infty} \left(\frac{1 + \frac{1}{n}}{2}\right)^n = \left(\frac{1}{2}\right)^\infty = 0.\]Since it doesn't approach 1, \( \limsup_{n \to \infty} \left(\frac{1 + \frac{1}{n}}{2}\right) < 1 \), the series converges absolutely by the root test.
03

Result on Convergence

The given series \( \sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)^{n}}{(2n)^{n}} \) converges absolutely because the corresponding non-alternating series also converges.

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Key Concepts

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

Alternating Series
An alternating series is a series in which the signs of its terms alternate between positive and negative. This type of series generally has a specific form:
  • The general term in an alternating series can be written as either \((-1)^n a_n\) or \((-1)^{n+1} a_n\), where \(`a_n`\) is a sequence of positive terms.
  • To determine its convergence, we often use the Alternating Series Test.
  • The series \(\sum_{n=1}^{\infty} (-1)^n a_n\) converges if two conditions are met:
    • \( a_{n+1} \leq a_n\) for all \(n\) (the terms are decreasing).
    • \( \lim_{n o \infty} a_n = 0\).
For example, in our exercise, the series \(\sum_{n=1}^{\infty} \frac{(-1)^n (n+1)^n}{(2n)^n}\) is an alternating series because of the factor \((-1)^n\). Understanding these concepts allows us to check if the alternating series converges based on the above conditions.
Absolute Convergence
Absolute convergence is a stronger form of convergence compared to regular convergence. A series \(\sum a_n\) converges absolutely if the series of absolute values \(\sum |a_n|\) also converges.
  • This concept helps determine whether a series will remain convergent when the absolute value is applied, removing any alternating signs.
  • The key advantage: if a series converges absolutely, it also converges conditionally, i.e., with its original signs intact.
  • The condition is mathematically written as: A series \(\sum a_n\) converges absolutely if \(\sum |a_n| = \sum \frac{(n+1)^n}{(2n)^n}\) converges.
In the given exercise, checking for absolute convergence involved considering the transformation of terms from the alternating series to a non-alternating one. By analyzing \(\sum \frac{(n+1)^n}{(2n)^n}\), we found it converged. Therefore, the original series converges absolutely, ensuring robust convergence.
Root Test
The root test, also known as Cauchy's root test, provides a convenient way to determine the convergence behavior of a series. This test involves:
  • Identifying the limit: Calculate \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\).
  • Compare this limit to 1 to decide convergence:
    • If the limit is less than 1, the series converges absolutely.
    • If the limit is greater than 1, the series diverges.
    • If the limit equals 1, the test is inconclusive.
In our exercise, the root test was applied to the series \(\sum \left(\frac{n+1}{2n}\right)^n\). Conducting this test entailed computing the root of each term and examining the limit as \(n\) approaches infinity. The result was that the computed limit was less than 1, confirming absolute convergence of the series.

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Most popular questions from this chapter

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{1}=1, \quad a_{n+1}=2 a_{n}-3$$

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{n}=\frac{n+1}{n}$$

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Pythagorean triples \(\quad\) A triple of positive integers \(a, b,\) and \(c\) is called a Pythagorean triple if \(a^{2}+b^{2}=c^{2} .\) Let \(a\) be an odd positive integer and let $$b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \quad \text { and } \quad c=\left\lceil\frac{a^{2}}{2}\right\rceil$$ be, respectively, the integer floor and ceiling for \(a^{2} / 2\). a. Show that \(a^{2}+b^{2}=c^{2} .\) (Hint: Let \(a=2 n+1\) and express \(b \text { and } c \text { in terms of } n .)\) b. By direct calculation, or by appealing to the accompanying figure, find $$\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{\left\lceil\frac{a^{2}}{2}\right\rceil}.$$

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