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Chapter 9: Problem 59

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. $$0.3+0.03+0.003+\cdots$$

Short Answer

Expert verified
Answer: We can conjecture that the value of the infinite series is \(\frac{1}{3}\).

Step by step solution

01

Identify the first term (a) and the common ratio (r)

The given infinite series is $$0.3+0.03+0.003+\cdots$$ The first term (a) is 0.3, and the common ratio (r) can be found by dividing a consecutive term by the previous one. $$\text{Common ratio } (r)= \frac{0.03}{0.3} = \frac{1}{10}$$
02

Find the first four terms of the sequence of partial sums (S1, S2, S3, S4)

To find the first four partial sums, we sum up the terms of the series up to the desired term. $$S_1 = 0.3$$ $$S_2 = 0.3 + 0.03$$ $$S_3 = 0.3 + 0.03 + 0.003$$ $$S_4 = 0.3 + 0.03 + 0.003 + 0.0003$$ Now, let's find the value of each of the partial sums. $$S_1 = 0.3$$ $$S_2 = 0.3 + 0.03 = 0.33$$ $$S_3 = 0.3 + 0.03 + 0.003 = 0.333$$ $$S_4 = 0.3 + 0.03 + 0.003 + 0.0003 = 0.3333$$
03

Conjecture about the value of the infinite series

After finding the first four terms of the sequence of partial sums, we can observe that the values are getting closer to \(\frac{1}{3}\). Since the common ratio is between -1 and 1, i.e., \(|r|<1\), the infinite sum converges to a limit. Using the formula for the sum of an infinite geometric series, we can find the value of the infinite series. $$S_{\infty} = \frac{a}{1 - r} = \frac{0.3}{1 - \frac{1}{10}} = \frac{0.3}{\frac{9}{10}} = \frac{1}{3}$$ So, the conjecture is that the value of the infinite series = \(\frac{1}{3}\).

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Most popular questions from this chapter

Series of squares Prove that if \(\sum a_{k}\) is a convergent series of positive terms, then the series \(\Sigma a_{k}^{2}\) also converges.

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.

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. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5, n=0,1,2, \dots$$

An early limit Working in the early 1600 s, the mathematicians Wallis, Pascal, and Fermat were attempting to determine the area of the region under the curve \(y=x^{p}\) between \(x=0\) and \(x=1\) where \(p\) is a positive integer. Using arguments that predated the Fundamental Theorem of Calculus, they were able to prove that $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1}$$ Use what you know about Riemann sums and integrals to verify this limit.

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd. } \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N,\) the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7,\) and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?
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