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Chapter 9: Problem 58

In your own words, state the difference between absolute and conditional convergence of an alternating series.

Short Answer

Expert verified
Absolute convergence means that a series remains convergent when absolute values are applied to its terms, while conditional convergence refers to a series being convergent, but becomes divergent when absolute values are applied to its terms.

Step by step solution

01

Define Absolute Convergence

Absolute convergence refers to the property of a series, whether its terms are all positive or not, the series are said to be absolutely convergent if the series formed by taking the absolute values of the original series is convergent.
02

Define Conditional Convergence

Conditional convergence refers to the attribute of a series whereby it's convergent, yet the series formed by taking the absolute values of its terms is not convergent. It implies that, while the original series does converge, it doesn't meet the stricter criteria of absolute convergence.
03

Explain the Difference

The main difference between the two scenarios lies in whether or not the series remains convergent when absolute values are applied to its terms. An absolutely convergent series remains convergent when absolute values are applied, whereas a conditionally convergent series becomes divergent when the same operation is executed.

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Most popular questions from this chapter

Finding a Series Explain how to use the series $$g(x)=e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$ to find the series for each function. Do not find the series. (a) \(f(x)=e^{-x}\) (b) \(f(x)=e^{3 x}\) (c) \(f(x)=x e^{x}\)

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