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Chapter 10: Problem 17

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum\left(\frac{x}{3}\right)^{k}$$

Short Answer

Expert verified
Answer: The radius of convergence is 3, and the interval of convergence is (-3, 3).

Step by step solution

01

Apply Ratio Test

The Ratio Test is a technique that calculates the limit of the absolute value of the ratio of consecutive terms of a series when k tends towards infinity. The Ratio Test states that a series converges if the limit L is less than 1, diverges if the limit L is greater than 1, and is inconclusive if the limit is equal to 1. For the given power series, we will determine the ratio first: $$\frac{a_{k+1}}{a_k} = \frac{\left(\frac{x}{3}\right)^{k+1}}{\left(\frac{x}{3}\right)^k} = \frac{x}{3}$$ Then, we will find the limit of this ratio as k tends to infinity: $$L = \lim_{k\to\infty} \left|\frac{x}{3}\right| = \frac{|x|}{3}$$
02

Determine Radius of Convergence

According to the Ratio Test, the series converges if L < 1: $$\frac{|x|}{3} < 1$$ Now, we will solve this inequality for x to find the radius of convergence: $$|x| < 3$$ The inequality states that the series converges when the absolute value of x is less than 3, which corresponds to a radius of convergence of 3.
03

Test the Endpoints

The inequality gives us the interval of convergence without including the endpoints: $$-3 < x < 3$$ Now, we need to test the endpoints to determine if they belong to the interval of convergence. We will analyze the given power series when x = -3 and x = 3: 1. x = -3: $$\sum\left(\frac{-3}{3}\right)^{k} = \sum\left(-1\right)^{k}$$ This is an alternating series, and since its terms do not approach 0 when k tends to infinity, it does not converge. 2. x = 3: $$\sum\left(\frac{3}{3}\right)^{k} = \sum 1^{k} = \sum 1$$ The power series elements do not approach 0, and the series is merely a sum of 1's, which diverges.
04

Determine the Interval of Convergence

As the given power series does not converge at the endpoints, the interval of convergence remains the same: $$-3 < x < 3$$ Therefore, the radius of convergence is 3, and the interval of convergence is (-3, 3).

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Most popular questions from this chapter

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\tan x$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)=6 y(t)+9, y(0)=2$$

Let $$f(x)=\sum_{k=0}^{\infty} c_{k} x^{k} \quad \text { and } \quad g(x)=\sum_{k=0}^{\infty} d_{k} x^{k}$$ a. Multiply the power series together as if they were polynomials, collecting all terms that are multiples of \(1, x,\) and \(x^{2} .\) Write the first three terms of the product \(f(x) g(x)\) b. Find a general expression for the coefficient of \(x^{n}\) in the product series, for \(n=0,1,2, \ldots\)

Errors in approximations Suppose you approximate \(\sin x\) at the points \(x=-0.2,-0.1,0.0,0.1,\) and 0.2 using the Taylor polynomials \(p_{3}=x-x^{3} / 6\) and \(p_{5}=x-x^{3} / 6+x^{5} / 120 .\) Assume that the exact value of \(\sin x\) is given by a calculator. a. Complete the table showing the absolute errors in the approximations at each point. Show two significant digits. $$\begin{array}{|c|l|l|} \hline x & \text { Error }=\left|\sin x-p_{3}(x)\right| & \text { Error }=\left|\sin x-p_{5}(x)\right| \\ \hline-0.2 & & \\ \hline-0.1 & & \\ \hline 0.0 & & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$ b. In each error column, how do the errors vary with \(x\) ? For what values of \(x\) are the errors the largest and smallest in magnitude?
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