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Chapter 10: Problem 17

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{3 \tan ^{-1} x-3 x+x^{3}}{x^{5}}$$

Short Answer

Expert verified
Answer: The limit of the given expression as x approaches 0 is 0.

Step by step solution

01

Find the Taylor Series of Inverse Tangent and Simplify the Expression

Recall that the Taylor series of a function f(x) around x = 0 is given by: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$ The Taylor series for the inverse tangent function, \(\tan^{-1}(x)\), is well established and is given by: $$\tan^{-1}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}$$ Therefore, the Taylor series for \(3\tan^{-1}(x)\) is: $$3\tan^{-1}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n 3}{2n+1} x^{2n+1}$$ Now let's simplify the given expression using this Taylor series: $$\frac{3 \tan ^{-1} x-3 x+x^{3}}{x^{5}} = \frac{\sum_{n=0}^{\infty} \frac{(-1)^n 3}{2n+1} x^{2n+1} - 3x + x^3}{x^5}$$
02

Combine Terms and Simplify the Expression Further

Now, let's combine terms and be careful to account for the different powers of x: $$\frac{\sum_{n=0}^{\infty} \frac{(-1)^n 3}{2n+1} x^{2n+1} - 3x + x^3}{x^5} = \frac{\left(\frac{3}{1}x - \frac{3}{3}x^3 + \frac{3}{5}x^5 - \dots \right) - 3x + x^3}{x^5}$$ After combining terms, we get: $$\frac{(-\frac{2}{3}x^3 + \frac{3}{5}x^5 - \dots)}{x^5}$$
03

Determine the Limit of the Simplified Expression as x Approaches 0

Now that we have our simplified expression, we can determine the limit as x approaches 0: $$\lim_{x \rightarrow 0} \frac{(-\frac{2}{3}x^3 + \frac{3}{5}x^5 - \dots)}{x^5}$$ Notice that every term in the numerator has a power of x greater than or equal to 3. This means that when we divide by \(x^5\), every term in the limit will be of the form \(\frac{constant}{x^k}\), where \(k\) is a positive even integer. As x approaches 0, every one of those terms approaches 0, and the limit becomes: $$\lim_{x \rightarrow 0} \frac{(-\frac{2}{3}x^3 + \frac{3}{5}x^5 - \dots)}{x^5} = 0$$ So, the limit of the given expression as x approaches 0 is 0.

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Most popular questions from this chapter

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$\sin 0.2$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The Taylor polynomials for \(f(x)=e^{-2 x}\) centered at 0 consist of even powers only. b. For \(f(x)=x^{5}-1,\) the Taylor polynomial of order 10 centered at \(x=0\) is \(f\) itself. c. The \(n\) th-order Taylor polynomial for \(f(x)=\sqrt{1+x^{2}}\) centered at 0 consists of even powers of \(x\) only.

The expected (average) number of tosses of a fair coin required to obtain the first head is \(\sum_{k=1}^{\infty} k\left(\frac{1}{2}\right)^{k} .\) Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)

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

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. $$f(x)=\sqrt{1-x^{2}}$$
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