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Chapter 8: Problem 2

Describe how to apply the Alternating Series Test.

Short Answer

Expert verified
Question: Explain how to apply the Alternating Series Test to determine the convergence or divergence of an alternating series. Answer: To apply the Alternating Series Test, follow these steps: 1. Identify an alternating series of the form \(\sum_{n=1}^{\infty} (-1)^n a_n\) or \(\sum_{n=1}^{\infty} (-1)^{n-1} a_n\), where the terms \(a_n\) are non-negative. 2. Ensure that the series' terms \(a_n\) are non-increasing and non-negative. 3. Apply the test by finding the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} a_n\). 4. Determine convergence or divergence: - If the limit is zero (i.e., \(\lim_{n \to \infty} a_n = 0\)), the series converges. - If the limit does not equal zero or does not exist, the series diverges.

Step by step solution

01

1. Identify the Alternating Series

An alternating series has a general form of: \(\sum_{n=1}^{\infty} (-1)^n a_n\) or \(\sum_{n=1}^{\infty} (-1)^{n-1} a_n\) where the terms \(a_n\) are non-negative. In mathematical notation, it can be described as \(a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + ... \)
02

2. Check the Series' Terms for Non-Increasing and Non-Negative

To apply the Alternating Series Test, the terms of the series \(a_n\) must be non-negative (i.e., \(a_n \ge 0\) for all \(n\)) and non-increasing (i.e., \(a_n \ge a_{n+1}\) for all \(n\)). Ensure that the two conditions are satisfied before applying the test to the series.
03

3. Apply the Alternating Series Test

Once the alternating series and its terms meet the conditions, apply the Alternating Series Test by taking the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} a_n\)
04

4. Determine Convergence or Divergence

Evaluate the limit obtained in step 3. - If the limit is zero, i.e., \(\lim_{n \to \infty} a_n = 0\), then the alternating series converges. - If the limit does not equal zero or does not exist, the alternating series diverges. After these steps, you can conclude whether the given alternating series converges or diverges based on the result of the Alternating Series Test.

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Most popular questions from this chapter

Find the limit of the sequence $$\left\\{a_{n}\right\\}_{n=2}^{\infty}=\left\\{\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \cdots\left(1-\frac{1}{n}\right)\right\\}_{n=2}^{\infty}.$$

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.

In \(1978,\) in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy: A family may have children until a boy is born. Suppose that the one-son policy were implemented and that natural birth rates remained the same (half boys and half girls). Using geometric series, compare the total number of children under the two policies.

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{3 n^{2}}{4 n^{2}+1}=\frac{3}{4}$$

Consider series \(S=\sum_{k=0}^{n} r^{k},\) where \(|r|<1\) and its sequence of partial sums \(S_{n}=\sum_{k=0}^{n} r^{k}\) a. Complete the following table showing the smallest value of \(n,\) calling it \(N(r),\) such that \(\left|S-S_{n}\right|<10^{-4},\) for various values of \(r .\) For example, with \(r=0.5\) and \(S=2,\) we find that \(\left|S-S_{13}\right|=1.2 \times 10^{-4}\) and \(\left|S-S_{14}\right|=6.1 \times 10^{-5}\) Therefore, \(N(0.5)=14\) $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline r & -0.9 & -0.7 & -0.5 & -0.2 & 0 & 0.2 & 0.5 & 0.7 & 0.9 \\ \hline N(r) & & & & & & & 14 & & \\ \hline \end{array}$$ b. Make a graph of \(N(r)\) for the values of \(r\) in part (a). c. How does the rate of convergence of the geometric series depend on \(r ?\)
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