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Chapter 10: Problem 56

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{3^{k}}$$

Short Answer

Expert verified
Answer: The function represented by the given power series is \(f(x) = \frac{3}{3+x}\).

Step by step solution

01

Recognize the power series

First, we notice that the given power series has the following form: $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{3^{k}}$$ This looks like a geometric series, with a common ratio \(r = -\frac{x}{3}\).
02

Apply the formula for the sum of a geometric series

The sum of an infinite geometric series can be found using the formula: $$S = \frac{a}{1-r}$$ Here, \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. In our case, \(a = \frac{x^{0}}{3^{0}} = 1\) and \(r = -\frac{x}{3}\).
03

Calculate the sum of the power series

Plugging values of \(a\) and \(r\) into the formula, we get the sum of the power series as: $$S = \frac{1}{1-(-\frac{x}{3})} = \frac{1}{1+\frac{x}{3}}$$
04

Simplify the sum

We can simplify the sum by multiplying the numerator and the denominator by 3 to get rid of the fraction in the denominator: $$S = \frac{1\times 3}{(1+\frac{x}{3})\times 3} = \frac{3}{3+x}$$ So, the function represented by the given power series is: $$f(x) = \frac{3}{3+x}$$

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Most popular questions from this chapter

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty}(-1)^{k} \frac{k x^{k+1}}{3^{k}}$$

The theory of optics gives rise to the two Fresnel integrals $$S(x)=\int_{0}^{x} \sin t^{2} d t \text { and } C(x)=\int_{0}^{x} \cos t^{2} d t$$ a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\) b. Expand \(\sin t^{2}\) and \(\cos t^{2}\) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for \(S\) and \(C\) c. Use the polynomials in part (b) to approximate \(S(0.05)\) and \(C(-0.25)\) d. How many terms of the Maclaurin series are required to approximate \(S(0.05)\) with an error no greater than \(10^{-4} ?\) e. How many terms of the Maclaurin series are required to approximate \(C(-0.25)\) with an error no greater than \(10^{-6} ?\)

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\frac{1}{\left(1+4 x^{2}\right)^{2}}$$

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

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int_{0}^{2} \frac{d x}{1-x},\) one could expand the integrand in a Taylor series and integrate term by term. b. To approximate \(\pi / 3,\) one could substitute \(x=\sqrt{3}\) into the Taylor series for \(\tan ^{-1} x\) c. \(\sum_{k=0}^{\infty} \frac{(\ln 2)^{k}}{k !}=2\)
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