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Chapter 9: Problem 32

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{(-1)^{n}}{n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the given sequence is 0.

Step by step solution

01

Identify the sequence

The given sequence is \(\left\\{\frac{(-1)^{n}}{n}\right\\}\). The sequence alternates in sign due to the \((-1)^n\) factor.
02

Determine the absolute value of the sequence

In order to determine whether the sequence converges absolutely, we should examine the absolute value of the sequence terms. The absolute value of the sequence is given by: $$\frac{|-1|^n}{n} = \frac{1}{n}$$
03

Examine if the sequence is decreasing in absolute value

Now, we need to check if the sequence with the absolute value is decreasing. This is easy, as \(\frac{1}{n}\) is clearly decreasing when \(n\) is an integer greater than or equal to 1 (since the denominator is increasing while the numerator stays constant, the sequence will decrease).
04

Examine the limit of the absolute value of the sequence

Next, we will find the limit of the absolute value of the sequence as \(n\) approaches infinity: $$\lim_{n\to\infty} \frac{1}{n} = 0$$ Since the limit of the absolute value of the sequence converges to 0, the original sequence converges absolutely.
05

Find the limit of the original sequence

Since the original sequence converges absolutely and is decreasing, we conclude that the limit of the original sequence is: $$\lim_{n\to\infty} \frac{(-1)^n}{n} = 0$$ Thus, the limit of the given sequence is 0.

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Most popular questions from this chapter

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k}$$

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

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 \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)

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. $$a_{n+1}=2 a_{n}\left(1-a_{n}\right) ; a_{0}=0.3, n=0,1,2, \dots$$

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.
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