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Chapter 9: Problem 25

Use the Root Test to determine whether the following series converge. $$1+\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{3}\right)^{3}+\left(\frac{1}{4}\right)^{4}+\dots$$

Short Answer

Expert verified
Question: Determine the convergence of the series \(\sum_{n=1}^{\infty} \left(\frac{1}{n}\right)^{n}\) using the Root Test. Answer: The series \(\sum_{n=1}^{\infty} \left(\frac{1}{n}\right)^{n}\) converges.

Step by step solution

01

Recall the Root Test

The Root Test states that given a series \(\sum_{n=1}^{\infty} a_n\), if \(\lim_{n\to\infty} \sqrt[n]{|a_n|} = L\), then: - If \(L < 1\), the series converges. - If \(L > 1\), the series diverges. - If \(L = 1\), the test is inconclusive. In our case, the series is given by \(a_n = \left(\frac{1}{n}\right)^{n}\).
02

Apply the Root Test

We need to find the limit \(\lim_{n\to\infty} \sqrt[n]{|a_n|}\) for the series. So, we have to find: $$\lim_{n\to\infty} \sqrt[n]{\left|\left(\frac{1}{n}\right)^{n}\right|}$$ Since the terms of the series are positive, we can drop the absolute value signs: $$\lim_{n\to\infty} \sqrt[n]{\left(\frac{1}{n}\right)^{n}}$$
03

Simplify the expression

We can simplify the above expression by using the property \((a^b)^c = a^{bc}\): $$\lim_{n\to\infty} \left(\frac{1}{n}\right)^{n\cdot\frac{1}{n}} = \lim_{n\to\infty} \left(\frac{1}{n}\right)$$
04

Calculate the limit

Now calculate the limit: $$\lim_{n\to\infty} \left(\frac{1}{n}\right) = 0$$ Since \(0\) is less than 1, the Root Test tells us that the series converges.

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Most popular questions from this chapter

The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the aggregate constant, is \(\prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}},\) where \(\prod_{n=0}^{N} a_{n}\) represents the product \(a_{0} \cdot a_{1} \cdots a_{N}\). This infinite product is the limit of the sequence $$\left\\{\prod_{n=0}^{0} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \cdot \prod_{n=0}^{1} \frac{2^{n}}{\sqrt{1+2^{2 n}}}, \prod_{n=0}^{2} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \ldots .\right\\}.$$ Estimate the value of the aggregate constant.

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty}\left(\frac{\ln k}{k}\right)^{p}$$

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\). When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots $$ Use estimation techniques to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{2 / 3}}$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{2^{k}}$$
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