/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 94 Suppose a function \(f\) is defi... [FREE SOLUTION] | 91影视

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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

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0}\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G\). a. Show that \(a_{n}>b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\). When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots $$ Use estimation techniques to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Define infinite series and give an example.

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$
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