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Chapter 10: Problem 46

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\sqrt{e}$$

Short Answer

Expert verified
Answer: The first four nonzero terms of the Taylor series for the given function are $$1 + \frac{1}{2}x + \frac{1}{8}x^2 + \frac{1}{48}x^3$$.

Step by step solution

01

Define the Function

Let's start by defining the function we want to approximate with a Taylor series: $$f(x) = \sqrt{e^{x}}$$
02

Find the first 4 derivatives of the function f(x)

Now, find the first four derivatives of the function f(x): 1. First Derivative: $$f'(x) = \frac{d}{dx}\sqrt{e^{x}} = \frac{1}{2}(e^{x})^{-\frac{1}{2}}(e^{x}) = \frac{1}{2}e^{\frac{x}{2}}$$ 2. Second Derivative: $$f''(x) = \frac{d}{dx}\left(\frac{1}{2}e^{\frac{x}{2}}\right) = \frac{1}{4}e^{\frac{x}{2}}$$ 3. Third Derivative: $$f^{(3)}(x) = \frac{d}{dx}\left(\frac{1}{4}e^{\frac{x}{2}}\right) = \frac{1}{8}e^{\frac{x}{2}}$$ 4. Fourth Derivative: $$f^{(4)}(x) = \frac{d}{dx}\left(\frac{1}{8}e^{\frac{x}{2}}\right) = \frac{1}{16}e^{\frac{x}{2}}$$
03

Evaluate the derivatives at x=0

Now, evaluate all the derivatives obtained in step 2 at x=0. 1. First Derivative: $$f'(0) = \frac{1}{2}e^{\frac{0}{2}} = \frac{1}{2}$$ 2. Second Derivative: $$f''(0) = \frac{1}{4}e^{\frac{0}{2}} = \frac{1}{4}$$ 3. Third Derivative: $$f^{(3)}(0) = \frac{1}{8}e^{\frac{0}{2}} = \frac{1}{8}$$ 4. Fourth Derivative: $$f^{(4)}(0) = \frac{1}{16}e^{\frac{0}{2}} = \frac{1}{16}$$
04

Construct the Taylor series

Now, let's write the Taylor series using the values obtained in step 3. $$\begin{aligned} f(x) &= f(0) + \frac{f'(0)}{1!}(x-0)^1 + \frac{f''(0)}{2!}(x-0)^2 + \frac{f^{(3)}(0)}{3!}(x-0)^3 + \cdots \\ &= e^0 + \frac{1}{2}x + \frac{1}{4}\frac{1}{2}x^2 + \frac{1}{8}\frac{1}{6}x^3 + \cdots \\ &= 1 + \frac{1}{2}x + \frac{1}{8}x^2 + \frac{1}{48}x^3 + \cdots \end{aligned}$$ The first four nonzero terms of the Taylor series of $$f(x) = \sqrt{e^x}$$ are: $$1 + \frac{1}{2}x + \frac{1}{8}x^2 + \frac{1}{48}x^3$$

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Most popular questions from this chapter

Suppose \(f\) and \(g\) have Taylor series about the point \(a.\) a. If \(f(a)=g(a)=0\) and \(g^{\prime}(a) \neq 0,\) evaluate \(\lim _{x \rightarrow a} f(x) / g(x).\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with l'H脙麓pital's Rule. b. If \(f(a)=g(a)=f^{\prime}(a)=g^{\prime}(a)=0\) and \(g^{\prime \prime}(a) \neq 0\) evaluate \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) by expanding \(f\) and \(g\) in their Taylor series. Show that the result is consistent with two applications of I'H脙麓pital's Rule.

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} \frac{x^{k}}{2^{k}}$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\cos 2$$

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