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Chapter 7: Problem 47

Give examples that show that the convergence of a power series at an endpoint of its interval of convergence may be either conditional or absolute. Explain your reasoning.

Short Answer

Expert verified
The alternating harmonic series \(危((-1)^n/n)\) provides an example of conditional convergence at an end point, because it converges, but does not converge when all terms are replaced by their absolute values. On the other hand, the series \(危x^n/n!\) is an example of absolute convergence because it does converge when all terms are replaced by their absolute values.

Step by step solution

01

Provide an Example of Conditional Convergence

Consider the alternating harmonic series \(-1^n/n\) for \(x = -1\), also known as \(危((-1)^n/n)\). This series converges conditionally at \(x = -1\) because its terms do not converge to 0.
02

Explain Convergence Test

The alternating series test states that if the terms decrease to zero, which they do in the above series as \(n\) approaches infinity, the series converges.
03

Provide an Example of Absolute Convergence

Now consider the series \(危x^n/n!\) for \(x = -1\) which equals \(危((-1)^n/n!)\). This series also converges at \(x = -1\) but unlike the above series, it converges absolutely.
04

Explain Absolute Convergence Test

The ratio test of convergence states if the absolute value of the ratio of the \(n+1\)th term to the nth term is less than 1, the series converges absolutely. This holds true for this series.

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Most popular questions from this chapter

Compound Interest Consider the sequence \(\left\\{A_{n}\right\\}\) whose \(n\) th term is given by \(A_{n}=P\left(1+\frac{r}{12}\right)^{n}\) where \(P\) is the principal, \(A_{n}\) is the account balance after \(n\) months, and \(r\) is the interest rate compounded annually. (a) Is \(\left\\{A_{n}\right\\}\) a convergent sequence? Explain. (b) Find the first 10 terms of the sequence if \(P=\$ 9000\) and \(r=0.055\)

Let \(\left\\{a_{n}\right\\}\) be a monotonic sequence such that \(a_{n} \leq 1\). Discuss the convergence of \(\left\\{a_{n}\right\\} .\) If \(\left\\{a_{n}\right\\}\) converges, what can you conclude about its limit?

Let \(\left\\{x_{n}\right\\}, n \geq 0,\) be a sequence of nonzero real numbers such that \(x_{n}^{2}-x_{n-1} x_{n+1}=1\) for \(n=1,2,3, \ldots .\) Prove that there exists a real number \(a\) such that \(x_{n+1}=a x_{n}-x_{n-1},\) for all \(n \geq 1 .\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(n>1\), then \(n !=n(n-1) !\)

Use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which each series converges. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}}, \quad \sum_{n=1}^{\infty}(0.01)^{n} $$
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