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Chapter 9: Problem 94

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Short Answer

Expert verified
Answer: The values of \(f(x)\) for the given values of \(x\) are: \(f(0) = 1\), \(f(0.2) = 1.25\), and \(f(0.5) = 2\). The function cannot be evaluated for \(x=1\) and \(x=1.5\) as the series does not converge for these values. The domain of the function \(f(x)\) is the open interval \((-1, 1)\).

Step by step solution

01

Sum of an infinite geometric series formula for f(x)

The sum of an infinite geometric series with common ratio r is given by the formula: \(S = \frac{a}{1-r}\) where \(a\) is the first term of the series. For our function \(f(x)\), the first term \(a=1\) and the common ratio is \(x\). So the sum formula for \(f(x)\) is: \(f(x) = \frac{1}{1-x}\)
02

Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible

Now, we'll use the formula found in Step 1 to evaluate f(x) for the given values of x: \(f(0) = \frac{1}{1-0} = 1\) \(f(0.2) = \frac{1}{1-0.2} = \frac{1}{0.8} = 1.25\) \(f(0.5) = \frac{1}{1-0.5} = \frac{1}{0.5} = 2\) For \(f(1)\) and \(f(1.5)\), the series does not converge, because the common ratio x is not between -1 and 1. Therefore, we cannot evaluate f(1) and f(1.5).
03

Find the domain of f(x)

The domain of \(f(x)\) is the set of all possible values of x for which the function f(x) is defined (i.e., the geometric series converges). As discussed earlier, the geometric series converges when the common ratio x is between -1 and 1: \(-1 < x < 1\) Therefore, the domain of \(f(x)\) is the open interval \((-1, 1)\).

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Most popular questions from this chapter

Consider the sequence \(\left\\{F_{n}\right\\}\) defined by $$F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)^{\prime}}$$ for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ). for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ).

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. A material transmutes \(50 \%\) of its mass to another element every 10 years due to radioactive decay. Let \(M_{n}\) be the mass of the radioactive material at the end of the \(n\) th decade, where the initial mass of the material is \(M_{0}=20 \mathrm{g}.\)

Find the limit of the sequence $$\left\\{a_{n}\right\\}_{n=2}^{\infty}=\left\\{\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \cdots\left(1-\frac{1}{n}\right)\right\\}.$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{2^{k}}$$
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