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Chapter 10: Problem 17

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=3^{x}$$

Short Answer

Expert verified
a) The first four nonzero terms of the Maclaurin series for the given function are: $$f(x) = 1 + (\ln(3))x + \frac{(\ln^2(3))x^2}{2!} + \frac{(\ln^3(3))x^3}{3!}$$ b) The power series can be written using summation notation as: $$f(x) = \sum_{n=0}^{\infty} \frac{(\ln^n(3))x^n}{n!}$$ c) The interval of convergence for this power series is: $$(-\infty, \infty)$$

Step by step solution

01

Find the first four nonzero terms of the Maclaurin series

First, obtain the functional derivatives. \(f'(x)=(3^x)'=\ln(3)*3^x\) \(f''(x)=(\ln(3)*3^x)'=\ln^2(3)*3^x\) \(f'''(x)=(\ln^2(3)*3^x)'=\ln^3(3)*3^x\) \(f''''(x)=(\ln^3(3)*3^x)'=\ln^4(3)*3^x\) Now, evaluate each derivative at x = 0 to get their respective coefficients: \(f(0)=3^0=1\) \(f'(0)=\ln(3)*3^0=\ln(3)\) \(f''(0)=\ln^2(3)*3^0=\ln^2(3)\) \(f'''(0)=\ln^3(3)*3^0=\ln^3(3)\) The first four nonzero terms of the Maclaurin series are: $$f(x) = 1 + (\ln(3))x + \frac{(\ln^2(3))x^2}{2!} + \frac{(\ln^3(3))x^3}{3!}$$
02

Write the power series using summation notation

We can write the power series as a summation: $$f(x) = \sum_{n=0}^{\infty} \frac{(\ln^n(3))x^n}{n!}$$
03

Determine the interval of convergence

To determine the interval of convergence, we will use the ratio test: $$\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \left(\frac{x\ln(3)}{n+1}\right)$$ The ratio test states that when this limit is less than 1, the series converges. Since this function is an exponential function, it will converge for all x: $$\lim_{n\to\infty} \left|\frac{x\ln(3)}{n+1}\right| < 1$$ So, the interval of convergence is: $$(-\infty, \infty)$$

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Most popular questions from this chapter

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=\sqrt{x} \text { with } a=36 ; \text { approximate } \sqrt{39}$$

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=1 / \sqrt{x} \text { with } a=4 ; \text { approximate } 1 / \sqrt{3}$$

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}$$

Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series. $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots, \text { for }-1 < x \leq 1$$ $$\sqrt{a^{2}+x^{2}}, a > 0$$

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.
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