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Chapter 8: Problem 17

Determine whether the following series converge. $$\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{2}}{k^{3}+1}$$

Short Answer

Expert verified
Answer: Yes, the given series converges.

Step by step solution

01

Identify the sequence terms

The given alternating series is: $$\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{2}}{k^{3}+1}$$ The sequence of terms can be identified as: $$a_k = \frac{k^2}{k^3 + 1}$$ Step 2: Verify if the sequence terms are positive and decreasing
02

Verify the sequence terms are positive and decreasing

To show that \(a_{k+1} \le a_{k}\), we must show that: $$\frac{k^2}{k^3 + 1} \ge \frac{(k+1)^2}{(k+1)^3 + 1}$$ After rearranging the terms, we get the inequality: $$(k+1)^3(k^2 + k^3 - k^2) \ge k^3(k^3 + k^2 - k^3)$$ Which simplifies to: $$(k+1)^3k^3 \ge k^3k^2$$ Since \(k^3\) is positive and can be canceled out, the inequality can be further simplified to: $$(k+1)^3 \ge k^2$$ This inequality holds true for all positive integers \(k\). Thus, the sequence satisfies the first property, since the sequence terms are positive and decreasing. Step 3: Verify if the sequence terms approach zero as \(k \rightarrow \infty\)
03

Verify if the sequence terms approach zero as \(k \rightarrow \infty\)

To show that \(a_k \to 0\) as \(k \to \infty\), we must show that: $$\lim_{k\to\infty} \frac{k^2}{k^3 + 1} = 0$$ Since \(k^3\) is the dominating term, it can be factored out in both the numerator and the denominator: $$\frac{k^2}{k^3 + 1} = \frac{k^3(1/k)}{k^3(1 + 1/k^3)} = \frac{1/k}{1 + 1/k^3}$$ As \(k\) goes to infinity, both \(1/k\) and \(1/k^3\) approach zero: $$\lim_{k\to\infty} \frac{1/k}{1 + 1/k^3} = \frac{0}{1 + 0} = 0$$ Thus, the sequence satisfies the second property, since the sequence terms approach zero as \(k \rightarrow \infty\). Step 4: Apply the Alternating Series Test
04

Apply the Alternating Series Test

As the sequence satisfies both the first and the second properties, the Alternating Series Test can be applied. Since all conditions are met, this implies that the series does converge. Answer: The given series converges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence of Series
When discussing the convergence of series, we refer to the behavior of an infinite sum when its terms are successively added. Specifically, a series converges if, as we add more terms, the sum approaches a finite, specific value. This is crucial for understanding functions and sequences within calculus and mathematical analysis.

An alternating series is one where the terms switch between positive and negative. The Alternating Series Test is a method we use to determine if these types of series converge. A series passes this test, and thus converges, if two conditions are met: the absolute value of the sequence terms is decreasing, and the limit of the sequence terms as the index approaches infinity is zero. As we can see in the exercise, the given series fulfills both criteria, and hence, we deduce that the series converges.
Sequence Terms
The term sequence terms refers to the individual elements of a sequence. A sequence is an ordered list of numbers, and each number in the sequence is called a 'term'. When considering the convergence of a series, especially when it is an alternating one, examining the behavior of these terms is essential.

In our example, the sequence terms are given by \(a_k = \frac{k^2}{k^3 + 1}\). To use the Alternating Series Test, we need to inspect these terms closely. They must be positive and decrease as \(k\) increases. As demonstrated in the solution steps, after performing the necessary comparisons, we can see that the terms indeed become smaller with each successive \(k\), complying with one of the test's conditions.
Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence get infinitely close to as the index goes to infinity. For a series to converge, particularly in an alternating series, it's imperative that the limit of the sequence terms is zero as the index (usually denoted as \(n\) or \(k\)) approaches infinity.

This concept is central to the Alternating Series Test. We can show that the limit is zero in our example by factoring out the dominating term \(k^3\) and simplifying the expression. Eventually, given \(\lim_{k\to\infty} \frac{k^2}{k^3 + 1} = 0\), we fulfill the second condition of the Alternating Series Test, confirming the series' convergence. It signifies that as the sequence progresses, the terms get so small that they essentially become negligible, hence ensuring the series won't head off towards infinity.

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Most popular questions from this chapter

Consider the alternating series $$ \sum_{k=1}^{\infty}(-1)^{k+1} a_{k}, \text { where } a_{k}=\left\\{\begin{array}{cl} \frac{4}{k+1}, & \text { if } k \text { is odd } \\ \frac{2}{k}, & \text { if } k \text { is even } \end{array}\right. $$ a. Write out the first ten terms of the series, group them in pairs, and show that the even partial sums of the series form the (divergent) harmonic series. b. Show that \(\lim _{k \rightarrow \infty} a_{k}=0\) c. Explain why the series diverges even though the terms of the series approach zero.

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month. At the end of each month, 120 fish are harvested. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. Assume that this process continues indefinitely. Use infinite series to find the longterm (steady-state) population of the fish.

Given any infinite series \(\sum a_{k}\) let \(N(r)\) be the number of terms of the series that must be summed to guarantee that the remainder is less than \(10^{-r}\) in magnitude, where \(r\) is a positive integer. a. Graph the function \(N(r)\) for the three alternating \(p\) -series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{p}},\) for \(p=1,2,\) and \(3 .\) Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) and compare the rates of convergence of all four series.

The sequence \(\\{n !\\}\) ultimately grows faster than the sequence \(\left\\{b^{n}\right\\},\) for any \(b>1,\) as \(n \rightarrow \infty .\) However, \(b^{n}\) is generally greater than \(n !\) for small values of \(n\). Use a calculator to determine the smallest value of \(n\) such that \(n !>b^{n}\) for each of the cases \(b=2, b=e,\) and \(b=10\).

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.
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