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Chapter 7: Problem 42

Use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1} $$

Short Answer

Expert verified
The series \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}\) diverges.

Step by step solution

01

Write down the given series

The given series is \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}\).
02

Identified a similar known series

The series \(\frac{1}{\sqrt{n}}\) or \(\sum_{n=2}^{\infty} \frac{1}{ \sqrt{n}}\) is identified. This series is known to diverge because it is a p-series with \(p = \frac{1}{2}\), which is less than 1.
03

Direct Comparison Test

Now we need to check the condition of the Direct Comparison Test, \(\frac{1}{\sqrt{n}-1} \geq \frac{1}{\sqrt{n}}\), for all \(n \geq 2\), we find this to be true, since the denominator of \(\frac{1}{\sqrt{n}-1}\) is less than that of \(\frac{1}{\sqrt{n}}\), ensuring that the value of \(\frac{1}{\sqrt{n}-1}\) is larger.
04

Conclusion

According to the Direct Comparison Test, if 0 鈮� a鈧� 鈮� b鈧� for all n in some range, and if the series \(\sum_{n=2}^{\infty} b鈧橽) diverges, then \(\sum_{n=2}^{\infty} a鈧橽) also diverges. Here, \(a鈧� = \frac{1}{\sqrt{n}-1}\) and \(b鈧� = \frac{1}{\sqrt{n}}\). Since \(\sum_{n=2}^{\infty} b鈧橽) diverges and \(a鈧� \geq b鈧橽) for all \(n \geq 2\), the given series \(\sum_{n=2}^{\infty} a鈧橽) also diverges.

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Most popular questions from this chapter

In Exercises 87 and 88 , use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=3\left[\frac{1-(0.5)^{x}}{1-0.5}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 3\left(\frac{1}{2}\right)^{n}} $$

Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{4^{n}} $$

Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\Sigma a_{n}\) also converges.

Consider the sequence \(\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}, n \geq 2\). (c) Find \(\lim _{n \rightarrow \infty} a_{n}\).

Modeling Data The annual sales \(a_{n}\) (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) represents the year, with \(n=3\) corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report) (3,3844),(4,4267),(5,4492),(6,4814),(7,5079) (8,5213),(9,5289),(10,5682),(11,5958),(12,6171) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=3,4, \ldots, 12\) for the data. Graphically compare the points and the model. (b) Use the model to predict sales in the year 2008 .
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