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Chapter 8: Problem 47

Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. $$\left\\{(-0.7)^{n}\right\\}$$

Short Answer

Expert verified
If it converges, what is the limit? Answer: The sequence converges, oscillates between positive and negative values, and has a limit of 0 when n approaches infinity.

Step by step solution

01

Observe the Sequence

First, let's observe what the sequence looks like by listing out the first few terms: $$ a_1=(-0.7)^1=-0.7\\ a_2=(-0.7)^2=0.49\\ a_3=(-0.7)^3=-0.343\\ a_4=(-0.7)^4=0.2401\\ $$ Observe that the sequence alternates in sign.
02

Check for Convergence

Since the terms alternate in sign, we can determine whether the sequence converges or diverges by using the Alternating Series Test. The alternating series test requires checking the limit of the absolute value of the sequence and whether it is decreasing. For our sequence, consider the absolute value $$|a_n|=\left|(-0.7)^n\right|=0.7^n$$. Let's find the limit of this. $$ \lim_{n\to\infty} 0.7^n $$ Since \(0.7 < 1\), we know that the limit converges to 0: $$ \lim_{n\to\infty} 0.7^n=0 $$ Now, let's check if the sequence $$|a_n|=0.7^n$$ is decreasing. We'll do this by considering consecutive terms and see whether the inequality holds: $$ 0.7^{n+1} \le 0.7^n \\ 0.7^n \cdot 0.7 \le 0.7^n $$ Since \(0.7 < 1\), the inequality holds, and the sequence is decreasing. Thus, by the alternating series test, we can conclude that the sequence $$\left\\{(-0.7)^{n}\right\\}$$ converges.
03

Identify Monotonic or Oscillating

As we observed in Step 1, the sequence alternates in sign, and this means it's oscillating between positive and negative values. Therefore, we can say that the sequence is oscillating.
04

Find the Limit

Since the sequence converges, we can now find the limit as $$n \to \infty$$. We observed earlier that the limit of the absolute value of the sequence converges to 0: $$ \lim_{n\to\infty} 0.7^n=0 $$ Thus, the limit of the original sequence is also 0: $$ \lim_{n\to\infty} (-0.7)^n=0 $$
05

Conclusion

The sequence $$\left\\{(-0.7)^{n}\right\\}$$ converges, oscillates between positive and negative values, and has a limit of 0 when n approaches infinity.

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Most popular questions from this chapter

Imagine that the government of a small community decides to give a total of \(\$ W\), distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction \(p\) of his or her new wealth and spends the remaining \(1-p\) in the community. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of \(\$ W\) increased (in terms of \(p\) )? Economists refer to this increase in the investment as the multiplier effect. b. Evaluate the limits \(p \rightarrow 0\) and \(p \rightarrow 1,\) and interpret their meanings.

The Fibonacci sequence \(\\{1,1,2,3,5,8,13, \ldots\\}\) is generated by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1},\) for \(n=1,2,3, \ldots,\) where \(f_{0}=1, f_{1}=1\). a. It can be shown that the sequence of ratios of successive terms of the sequence \(\left\\{\frac{f_{n+1}}{f_{n}}\right\\}\) has a limit \(\varphi .\) Divide both sides of the recurrence relation by \(f_{n},\) take the limit as \(n \rightarrow \infty,\) and show that \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}}=\frac{1+\sqrt{5}}{2} \approx 1.618\). b. Show that \(\lim _{n \rightarrow \infty} \frac{f_{n-1}}{f_{n+1}}=1-\frac{1}{\varphi} \approx 0.382\). c. Now consider the harmonic series and group terms as follows: $$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\left(\frac{1}{4}+\frac{1}{5}\right)+\left(\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)$$ $$+\left(\frac{1}{9}+\cdots+\frac{1}{13}\right)+\cdots$$ With the Fibonacci sequence in mind, show that $$\sum_{k=1}^{\infty} \frac{1}{k} \geq 1+\frac{1}{2}+\frac{1}{3}+\frac{2}{5}+\frac{3}{8}+\frac{5}{13}+\cdots=1+\sum_{k=1}^{\infty} \frac{f_{k-1}}{f_{k+1}}.$$ d. Use part (b) to conclude that the harmonic series diverges. (Source: The College Mathematics Journal, 43, May 2012)

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

The expression $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+}}}}.$$ 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 \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. 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 $$a+\frac{b}{a+\frac{b}{a+\frac{b}{a+\frac{b}{a+}}}}$$ where \(a\) and \(b\) are positive real numbers.

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}.$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\). b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\). c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\).
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