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Chapter 8: Problem 14

Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}$$

Short Answer

Expert verified
Question: Determine if the series $$\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}$$ converges or diverges using the Divergence Test. Answer: The series $$\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}$$ diverges.

Step by step solution

01

Write the given series

Our given series is: $$\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}$$
02

Calculate the limit of the terms

Find the limit of the terms as k approaches infinity: $$\lim_{k\to\infty} \frac{k^{3}}{k^{3}+1}$$
03

Simplify the limit expression

Divide both the numerator and the denominator of the limit expression by the highest power of k, which is \(k^3\): $$\lim_{k\to\infty} \frac{k^{3}/k^3}{(k^{3}+1)/k^{3}} = \lim_{k\to\infty} \frac{1}{1+\frac{1}{k^3}}$$
04

Evaluate the limit

Now, evaluate the limit as k approaches infinity: $$\lim_{k\to\infty} \frac{1}{1+\frac{1}{k^3}} = \frac{1}{1+0} = 1$$
05

Apply the Divergence Test

The limit of the terms is not zero (it equals 1), so the Divergence Test tells us that the series diverges. Therefore, the series $$\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}$$ diverges.

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Key Concepts

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

Series
A series is a sum of terms of a sequence. It is expressed in mathematics using the summation symbol, often denoted with \( \sum \). A series can be finite or infinite, depending on the number of terms. Finite series end after a specified number of terms, while infinite series continue indefinitely. In calculus and mathematical analysis, an infinite series is a sequence of numbers in which the terms are added indefinitely. The primary focus is often on whether an infinite series converges to a finite value or diverges, which can be determined by various tests and methods.
Divergence
Divergence refers to the behavior of a series that does not approach a finite limit as more terms are added. When we say that a series diverges, it means that as more terms are summed, the total grows infinitely or fails to stabilize around a specific value. The Divergence Test, also known as the nth-term test, is a simple method for testing whether a series diverges. The test states that if the limit of the terms does not equal zero, then the series must diverge. However, if the limit equals zero, the test is inconclusive and doesn't necessarily mean the series converges.
Convergence
Convergence is the condition where the sum of the series approaches a finite value as the number of terms increases. For a series to converge, the individual terms must become smaller and approach zero. Several methods help determine convergence, such as the Ratio Test, Root Test, and Integral Test, each applicable in different circumstances. It's essential to choose the right test for the series you are evaluating, as some tests more effectively address specific forms of series. Understanding convergence often requires analyzing the behavior of the sequence that forms the series.
Limit Evaluation
Limit evaluation involves determining the behavior of a function or sequence as its input approaches a particular value or infinity. This is a crucial process in calculus and is fundamental in series analysis. To evaluate the limit of a function, we use various techniques like factoring, multiplying by conjugates, or simplifying the expression by removing indeterminate forms. In the context of series, evaluating the limit of terms is necessary for applying convergence and divergence tests. For example, if the terms of a series do not tend to zero, the series cannot converge. Thus, mastery of limit evaluation is critical for analyzing series behavior.

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Most popular questions from this chapter

It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots=\ln 2 $$ Show that by rearranging the terms (so the sign pattern is \(++-\) ), $$ 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\cdots=\frac{3}{2} \ln 2 $$

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{1000}}{2^{n}}\right\\}$$

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{(-2)^{k}}{3^{k+1}}$$

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction \(p\) of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.
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