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Chapter 8: Problem 45

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. $$\left\\{0.2^{n}\right\\}$$

Short Answer

Expert verified
Question: Determine whether the sequence $$\left\\{0.2^{n}\right\\}$$ converges or diverges and describe its behavior. Answer: The sequence $$\left\\{0.2^{n}\right\\}$$ converges to 0 and is monotonic (decreasing).

Step by step solution

01

Understand the Sequence

The given sequence is: $$\left\\{0.2^{n}\right\\}$$ which can also be written as: $$\left\\{\left(\frac{1}{5}\right)^{n}\right\\}$$
02

Determine the limit

The limit of the sequence, if it exists, can be found using the following limit formula: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(\frac{1}{5}\right)^{n}$$ Since the ratio is between -1 and 1, the limit converges to 0: $$\lim_{n \to \infty} \left(\frac{1}{5}\right)^{n} = 0$$ The sequence converges to 0.
03

Determine monotonicity or oscillation

To determine if the sequence is monotonic or oscillating, we need to look at the progression. We can observe that: $$a_1 = 0.2$$ $$a_2 = 0.2^2 = 0.04$$ $$a_3 = 0.2^3 = 0.008$$ $$\cdots$$ From the above terms, we can see that the sequence is decreasing and has no oscillation. Therefore, the sequence is monotonic.
04

Conclusion

The given sequence $$\left\\{\left(\frac{1}{5}\right)^{n}\right\\}$$ converges to 0 and is monotonic (decreasing).

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Most popular questions from this chapter

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.

Suppose that you take 200 mg of an antibiotic every 6 hr. The half-life of the drug is 6 hr (the time it takes for half of the drug to be eliminated from your blood). Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood.

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Marie takes out a \(\$ 20,000\) loan for a new car. The loan has an annual interest rate of \(6 \%\) or, equivalently, a monthly interest rate of \(0.5 \% .\) Each month, the bank adds interest to the loan balance (the interest is always \(0.5 \%\) of the current balance), and then Marie makes a \(\$ 200\) payment to reduce the loan balance. Let \(B_{n}\) be the loan balance immediately after the \(n\) th payment, where \(B_{0}=\$ 20,000\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. Determine how many months are needed to reduce the loan balance to zero.

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