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Chapter 3: Problem 9

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n}}\) always convergent? Either prove it or give a counterexample.

Short Answer

Expert verified
The statement is false. A counterexample is the series \(a_{n} = \frac{1}{n^2}\) (which is convergent) and its square root series \(\sum \sqrt{a_{n}} = \sum {\frac{1}{n}}\) which is divergent.

Step by step solution

01

Understanding the problem statement

We have a series \(\sum a_{n}\) which we know is convergent, i.e., its partial sums form a sequence that has a finite limit. The problem asks if the series \(\sum \sqrt{a_{n}}\) is always convergent given this.
02

Formulating a counterexample

Consider a series whose terms are \(a_{n} = \frac{1}{n^2}\). We know this series to be convergent as it is a p-series with p=2 which is greater than 1. Now, consider \(\sum \sqrt{a_{n}} = \sum {\frac{1}{n}}\). This is a p-series with p=1 and is known to be divergent (harmonic series). This serves as a counterexample, disproving the statement that \(\sum \sqrt{a_{n}}\) is always convergent if \(\sum a_{n}\) is.
03

Final Conclusion

The series \(\sum \sqrt{a_{n}}\) is not always convergent even if the original series \(\sum a_{n}\) is convergent. A counterexample exists to disprove the claim (for instance, the series of \(\frac{1}{n^2}\) and its square root series). Therefore, convergence of an original series does not guarantee the convergence of its corresponding square root series.

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Most popular questions from this chapter

Let \(x_{n}:=1 / \ln (n+1)\) for \(n \in \mathbb{N}\) (a) Use the definition of limit to show that \(\lim \left(x_{n}\right)=0\). (b) Find a specific value of \(K(\varepsilon)\) as required in the definition of limit for each of (i) \(\varepsilon=1 / 2\), and (ii) \(\varepsilon=1 / 10\).

Find the limits of the following sequences: (b) \(\lim \left(\frac{(-1)^{n}}{n+2}\right)\), (a) \(\lim \left((2+1 / n)^{2}\right)\), (c) \(\lim \left(\frac{\sqrt{n}-1}{\sqrt{n}+1}\right)\). (d) \(\lim \left(\frac{n+1}{n \sqrt{n}}\right)\).

Suppose that every subsequence of \(X=\left(x_{n}\right)\) has a subsequence that converges to \(0 .\) Show that \(\lim X=0\)

Use the Cauchy Condensation Test to establish the divergence of the series: (a) \(\sum \frac{1}{n \ln n}\), (b) \(\sum \frac{1}{n(\ln n)(\ln \ln n)}\) (c) \(\sum \frac{1}{n(\ln n)(\ln \ln n)(\ln \ln \ln n)}\).

Let \(b \in \mathbb{R}\) satisfy \(0
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