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Chapter 10: Problem 43

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

Short Answer

Expert verified
The series converges, but not absolutely.

Step by step solution

01

Analyze the Series

The given series is \( \sum_{n=1}^{\infty} (-1)^{n} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \). Notice that it is an alternating series since \((-1)^n\) alternates signs as \(n\) increases.
02

Simplify the Expression

Consider the expression \( \sqrt{n+\sqrt{n}} - \sqrt{n} \). To simplify, multiply and divide by the conjugate: \[\sqrt{n+\sqrt{n}} - \sqrt{n} = \frac{n+\sqrt{n}-n}{\sqrt{n+\sqrt{n}}+\sqrt{n}} = \frac{\sqrt{n}}{\sqrt{n+\sqrt{n}}+\sqrt{n}}.\]
03

Analyze the Simplified Form

The simplified form \( \frac{\sqrt{n}}{\sqrt{n+\sqrt{n}}+\sqrt{n}} \) approaches 0 as \( n \to \infty \) because the denominator \( \sqrt{n+\sqrt{n}} + \sqrt{n} \to 2\sqrt{n} \), thus making the fraction approach \( \frac{1}{2} \).
04

Check Absolute Convergence

To check for absolute convergence, consider the series \( \sum_{n=1}^{\infty} \left|(-1)^{n} ( \sqrt{n+\sqrt{n}}-\sqrt{n})\right| = \sum_{n=1}^{\infty} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \). Since \( \sum_{n=1}^{\infty} \frac{1}{2} \) diverges, the series does not converge absolutely.
05

Apply Alternating Series Test

Use the Alternating Series Test to determine if the series converges: the terms \( b_n = \sqrt{n+\sqrt{n}} - \sqrt{n} \) satisfy \( b_n \to 0 \) as \( n \to \infty \), and \( b_{n+1} < b_n \) for sufficiently large \( n \). Thus, the series converges by the Alternating Series Test.

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Key Concepts

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

Absolute Convergence
When we talk about absolute convergence, we're looking at whether a series still converges when we take the absolute value of each term. In simpler terms, can the series indeed sum up to a finite limit even if we ignore the signs of its terms?
To determine this, examine the original series' absolute value:
  • For our series, that means analyzing \( \sum_{n=1}^{\infty} \left|(-1)^{n} ( \sqrt{n+\sqrt{n}}-\sqrt{n})\right| \).
  • When considering the absolute values, this expression simplifies to \( \sum_{n=1}^{\infty} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \).
  • Since this is similar to \( \sum_{n=1}^{\infty} \frac{1}{2} \), which is known to diverge, our series does not converge absolutely.
Series Convergence
Series convergence considers whether the entire series adds up to a finite value. When dealing with alternating series, like our original series \( \sum_{n=1}^{\infty} (-1)^{n} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \), we use specific tests designed for series of this nature.
One such method is the Alternating Series Test, which can confirm whether an alternating series converges:
  • It requires that \( b_n \to 0 \) as \( n \to \infty \).
  • It also requires that the terms decrease in size, so \( b_{n+1} < b_n \) eventually.
In our case, the terms do satisfy both conditions, proving that the series converges.
Divergence
A series diverges if it does not sum to a finite limit. This occurs when a series grows without bound or oscillates indefinitely without settling.
In our examination, the absolute series we derived, \( \sum_{n=1}^{\infty} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \), was similar to a divergent series \( \sum_{n=1}^{\infty} \frac{1}{2} \).
  • Since each term "increases" or does not diminish suitably fast between terms, the series diverges.
  • This insight helps us conclude the importance of analyzing not just position or operation, but the actual numeric consequence of long-term behavior.
Mathematical Analysis
Mathematical analysis provides the tools and frameworks for understanding limiting behaviors and convergence of sequences and series.
Using these techniques, we can examine whether a series like \( \sum_{n=1}^{\infty} (-1)^{n} ( \sqrt{n+\sqrt{n}}-\sqrt{n}) \) converges:
  • Analysis allowed for proper variable reductions and simplification, turning complex expressions into workable forms.
  • By interpreting bounds and limits, we could apply theorems such as the Alternating Series Test accurately.
Particularly, the concepts of convergence and divergence, backed by analytical methods, ensure our results are robust and reliable.

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Most popular questions from this chapter

Use the definition of \(e^{i \theta}\) to show that for any real numbers \(\theta, \theta_{1}\) and \(\theta_{2}\) a. \(\quad e^{i \theta_{1}} e^{i \theta_{2}}=e^{i\left(\theta_{1}+\theta_{2}\right)}\) b. \(e^{-i \theta}=1 / e^{i \theta}\)

a. Find the interval of convergence of the power series $$\sum_{n=0}^{\infty} \frac{8}{4^{n+2}} x^{n}$$ b. Represent the power series in part (a) as a power series about \(x=3\) and identify the interval of convergence of the new series. (Later in the chapter you will understand why the new interval of convergence does not necessarily include all of the numbers in the original interval of convergence.)

Which of the sequences converge, and which diverge? Give reasons for your answers. \(a_{n}=\frac{1+\sqrt{2 n}}{\sqrt{n}}\)

Determine if the sequence is monotonic and if it is bounded. \(a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}}\)

Two complex numbers \(a+i b\) and \(c+\) id are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate $$ \int e^{a x} \cos b x d x \quad \text { and } \quad \int e^{a x} \sin b x d x $$ from $$ \int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C $$ where \(C=C_{1}+i C_{2}\) is a complex constant of integration.
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