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Chapter 9: Problem 55

Define an alternating series.

Short Answer

Expert verified
An alternating series is a series of terms that alternate in sign. It can be defined as an infinite series of the form \[(-1)^n a_n\] or \[(-1)^{n+1} a_n\], where \(a_n\) is a sequence of positive numbers. An example is the alternating harmonic series: \(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots \).

Step by step solution

01

Understanding an Alternating Series

In mathematics, a series is the sum of the terms of a sequence. A sequence is a list of numbers, and a series is the sum of these numbers. A series can be finite or infinite. Now, in an alternating series, as the name suggests, the signs of the terms change alternatively. That means, if a term in the series is positive, then the next term will be negative, and the term following that will again be positive and so on.
02

Mathematical Definition of an Alternating Series

An alternating series can be formally defined as an infinite series whose terms alternate in sign. More specifically, an alternating series is a series of the form \[(-1)^n a_n\] or \[(-1)^{n+1} a_n\] where \(a_n\) is a sequence of positive numbers.
03

Discussing Example of Alternating Series

A common example of an alternating series is the alternating harmonic series: \(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots \). Here, as expected, we see that the signs of the terms are changing alternatively.

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