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Chapter 3: Problem 5

Can you give an example of a convergent series \(\sum x_{n}\) and a divergent series \(\sum y_{n}\) such that \(\sum\left(x_{n}+y_{n}\right)\) is convergent? Explain.

Short Answer

Expert verified
The harmonic series \(x_n = 1/n\) is divergent and the series \(y_n = -1/n\) is convergent. When they are added together, they result in a convergent series where all terms are zero. Therefore, \(\sum_{n=1}^\infty (x_n + y_n) = 0.\)

Step by step solution

01

Define the Series

First, let's define the series. Let \(x_n = 1/n\) for all \(n \geq 1\). This is a well-known series, called the harmonic series, which is divergent. Now let \(y_n = -1/n\) for all \(n \geq 1\). This series is convergent, because it adds up to zero.
02

Add the Series

In this step, let's add those two together to determine if the sum of the two, i.e., \(\sum_{n=1}^\infty (x_n + y_n) = \sum_{n=1}^\infty (1/n - 1/n)\), is convergent or not.
03

Evaluate the Result

In the last step, we see that \(\sum_{n=1}^\infty (1/n - 1/n) = \sum_{n=1}^\infty 0\), which is equal to zero for every term. Thus, the series converges, namely to the sum of all terms, which is zero.

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