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Chapter 9: Problem 73

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. $$\sum_{k=1}^{\infty} 3^{-k}$$

Short Answer

Expert verified
Answer: The estimated limit of the sequence of partial sums for the given infinite series is $\frac{1}{2}$.

Step by step solution

01

Write the first four terms of the sequence of partial sums

In order to find the partial sums, we need to sum the first \(n\) terms of the series for \(n=1,2,3,4\). Let's find the first four partial sums \(S_1, S_2, S_3\) and \(S_4\): 1. \(S_1 = 3^{-1} = \frac{1}{3}\) 2. \(S_2 = 3^{-1} + 3^{-2} = \frac{1}{3} + \frac{1}{9} = \frac{4}{9}\) 3. \(S_3 = 3^{-1} + 3^{-2} + 3^{-3} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{13}{27}\) 4. \(S_4 = 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{40}{81}\)
02

Estimate the limit or state that it does not exist

This is a geometric series with a common ratio \(r = \frac{1}{3}\). If \(|r| < 1\), the geometric series converges, and we can use the formula for the sum of an infinite geometric series: $$S_\infty = \frac{a_1}{1-r}$$ Here, the first term \(a_1 = 3^{-1} = \frac{1}{3}\). Since \(|r| = \frac{1}{3} < 1\), the series converges. Now, let's estimate the limit \(S_\infty\): $$S_\infty = \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}$$ So, the limit of the sequence of partial sums is \(\frac{1}{2}\).

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Most popular questions from this chapter

For a positive real number \(p,\) how do you interpret \(p^{p^{p \cdot *}},\) where the tower of exponents continues indefinitely? As it stands, the expression is ambiguous. The tower could be built from the top or from the bottom; that is, it could be evaluated by the recurrence relations \(a_{n+1}=p^{a_{n}}\) (building from the bottom) or \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{0}=p\) in either case. The two recurrence relations have very different behaviors that depend on the value of \(p\). a. Use computations with various values of \(p > 0\) to find the values of \(p\) such that the sequence defined by (2) has a limit. Estimate the maximum value of \(p\) for which the sequence has a limit. b. Show that the sequence defined by (1) has a limit for certain values of \(p\). Make a table showing the approximate value of the tower for various values of \(p .\) Estimate the maximum value of \(p\) for which the sequence has a limit.

Evaluate the limit of the following sequences. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{2 k+1}$$

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 \(0 . \overline{9}=1\)

Consider the series \(\sum_{k=3}^{\infty} \frac{1}{k \ln k(\ln \ln k)^{p}},\) where \(p\) is a real number. a. For what values of \(p\) does this series converge? b. Which of the following series converges faster? Explain. $$ \sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{2}} \text { or } \sum_{k=3}^{\infty} \frac{1}{k \ln k(\ln \ln k)^{2}} ? $$
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