/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 Find the limit of the following ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 36

Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. $$a_{n}=\frac{(-1)^{n} n}{n+1}$$

Short Answer

Expert verified
Answer: The sequence diverges.

Step by step solution

01

Analyze the Sequence

Break down the given sequence \(a_{n}=\frac{(-1)^{n} n}{n+1}\). We can rewrite the sequence as \(a_{n}=(-1)^{n}\frac{n}{n+1}\) to make it more understandable. Notice that \((-1)^{n}\) oscillates between -1 and 1 depending on whether n is even or odd, and that the fraction \(\frac{n}{n+1}\) is always between 0 and 1.
02

Determine Whether the Sequence Converges or Diverges

As n approaches infinity (\(n \to \infty\)), let's examine how each part of the sequence behaves. 1. \((-1)^{n}\): This part will oscillate between -1 and 1 forever, as n increases. 2. \(\frac{n}{n+1}\): As n gets infinitely large, this fraction will approach 1. These two pieces of information can help us to determine whether the sequence converges or diverges.
03

Examine the Product of the Sequence's Components

Because \((-1)^{n}\) oscillates between -1 and 1 and \(\frac{n}{n+1}\) approaches 1, their product (\(a_n\)) will also oscillate between positive and negative values as n approaches infinity. As a result, there is no limiting value that the sequence approaches, and we can conclude that the sequence diverges.
04

Check Your Result with a Graphing Utility

To confirm the results, graph the sequence using a graphing utility such as Desmos or GeoGebra. By plotting the sequence, you will observe that the terms of the sequence continue to oscillate between positive and negative values as the sequence progresses. This confirms that the sequence does not converge to a specific value. In conclusion, the limit of the given sequence does not exist as the sequence diverges.

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Most popular questions from this chapter

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1}\), for \(n=1,2,3, \ldots,\) where \(f_{0}=0, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$

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. $$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$ \ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n $$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$ \frac{1}{n+1}>\ln (n+2)-\ln (n+1) $$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\) e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\},\) estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}$$

In Section 3, we established that the geometric series \(\Sigma r^{k}\) converges provided \(|r|<1\). Notice that if \(-1
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