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Chapter 8: Problem 15

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

Short Answer

Expert verified
Answer: The given series converges.

Step by step solution

01

Identify the series as an alternating series

The given series is: $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3}}$$ Observe that the terms of the series have alternating signs due to the \((-1)^{k+1}\) factor. Thus, it is an alternating series.
02

Define the sequence of absolute values of the terms

Now, let's define the sequence of absolute values of the terms as \(a_k = \frac{1}{k^3}\). This sequence represents only the magnitude of the terms of the given series.
03

Show the sequence of absolute values is decreasing

We need to prove that the sequence \(a_k=\frac{1}{k^3}\) is decreasing. To show this, we can consider the ratio between consecutive terms: $$\frac{a_{k+1}}{a_k}=\frac{\frac{1}{(k+1)^3}}{\frac{1}{k^3}}=\frac{k^3}{(k+1)^3}$$ As \(k < k+1\), then \(k^3<(k+1)^3\). Therefore, \(\frac{k^3}{(k+1)^3} < 1\), which implies that the sequence \(a_k=\frac{1}{k^3}\) is decreasing.
04

Find the limit of the absolute values

Now, we need to find the limit of the absolute values as k approaches infinity: $$\lim_{k\to\infty}\frac{1}{k^{3}}$$ As \(k\) tends to infinity, \(k^3\) will also tend to infinity, and the fraction will tend to zero: $$\lim_{k\to\infty}\frac{1}{k^{3}}=0$$
05

Apply the Alternating Series Test

According to the Alternating Series Test, if the sequence of absolute values of an alternating series is decreasing and the limit of the absolute values as k approaches infinity is zero, then the series converges. Since we have shown that both conditions are true for our given series, we can conclude that the series converges. Therefore, the given series $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3}}$$ 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 we talk about the convergence of a series, we essentially want to know whether adding up all the terms of an infinite series will result in a finite sum. Not all series converge; some will keep growing indefinitely. But for convergent series, as we add more and more terms, the total sum approaches a specific number.
  • If a series converges, its terms must approach zero as you include more in the series.
  • Convergent series can sometimes be very subtle; they might start with large numbers but eventually add up to a finite number.
  • The series must follow a specific pattern or rule that allows us to apply mathematical tests to determine convergence.
Understanding convergence is crucial because it helps mathematicians and scientists predict and analyze behaviors over an infinite number of steps or periods. In our given example, by applying certain tests and verifying conditions, we establish that the series converges.
Alternating Series
An alternating series is a series where the signs of the terms are alternately positive and negative. This generally happens due to the presence of a factor like \((-1)^k\), which flips the sign of each term:
  • The pattern of changing signs is critical; it affects whether a series converges or diverges.
  • The Alternating Series Test is a crucial tool to check convergence here.
  • For a series to pass the Alternating Series Test, the absolute values of the terms must be decreasing, and the limit of these absolute values should approach zero.
In our series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3}}\), the construction \((-1)^{k+1}\) ensures that the series is indeed alternating. The importance lies in the behavior of the terms as the series progresses, ensuring both conditions of the Alternating Series Test are met.
Infinite Series
Infinite series can be confusing at first, but simply put, they are series that continue indefinitely. They consist of a potentially limitless sequence of numbers added together.
  • An infinite series is expressed mathematically with a summation symbol, indicating the sum of terms from a starting index to infinity.
  • Such series can converge to a finite value, or diverge, meaning they do not reach a finite value.
  • There are various tests and methods, like Comparison Test, Ratio Test, and the Alternating Series Test, to check for convergence.
The beauty of infinite series lies in their ability to model real-world phenomena where processes or changes are continuous. For our exercise, the series\[ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3}} \]is an infinite series that we've proven to converge. This shows how infinite series can be not only abstract but also have practical implications, illustrating order within what initially appears as endless chaos.

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Most popular questions from this chapter

Consider the number \(0.555555 \ldots,\) which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 .\) b. Consider the number \(0.54545454 \ldots\), which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots ., n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form for \(0 . \overline{n_{1}} n_{2} \cdots n_{p}\) d. Try the method of part (c) on the number \(0 . \overline{123456789}=0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

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

Assume that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\) (Exercises 65 and 66 ) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.

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.

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=n^{10} \text { and } b_{n}=n^{9} \ln ^{3} n, n \geq 7$$
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