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Chapter 9: Problem 60

For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. $$0.6+0.06+0.006+\cdots$$

Short Answer

Expert verified
Answer: The value of the infinite geometric series is \(\frac{2}{3}\).

Step by step solution

01

Identify the common ratio and the first term

We are given an infinite geometric series: $$0.6+0.06+0.006+\cdots$$ The first term, denoted by \(a\), is 0.6. The common ratio, denoted by \(r\), can be found by dividing the second term by the first term: $$r = \frac{0.06}{0.6} = \frac{1}{10}$$.
02

Find the first four partial sums

To find the first four terms of the sequence of partial sums, we can simply compute the sums: 1. \(S_1 = 0.6\) 2. \(S_2 = 0.6 + 0.06 = 0.66\) 3. \(S_3 = 0.66 + 0.006 = 0.666\) 4. \(S_4 = 0.666 + 0.0006 = 0.6666\) So, the first four partial sums are 0.6, 0.66, 0.666, and 0.6666.
03

Make a conjecture about the value of the infinite series

To make a conjecture about the value of the infinite series, let's first check if the series converges by examining the common ratio \(r\). Since \(|r| = |\frac{1}{10}|<1\), the series converges. Now we can use the formula for the sum of an infinite geometric series, which is: $$S_\infty = \frac{a}{1 - r}$$ Substitute the given values for \(a\) and \(r\): $$S_\infty = \frac{0.6}{1 - \frac{1}{10}} = \frac{0.6}{\frac{9}{10}}$$ Finally, compute the sum: $$S_\infty = \frac{0.6 * 10}{9} = \frac{6}{9} = \frac{2}{3}$$ Based on the calculations, it can be conjectured that the value of the infinite series is \(\frac{2}{3}\).

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Most popular questions from this chapter

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. $$a_{n+1}=\frac{1}{2}\left(a_{n}+2 / a_{n}\right) ; a_{0}=2, n=0,1,2, \dots$$

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. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{2 k+1}$$

For what values of \(x\) does the geometric series $$f(x)=\sum_{k=0}^{\infty}\left(\frac{1}{1+x}\right)^{k}$$ converge? Solve \(f(x)=3\)

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \cos (\pi k)$$
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