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Chapter 10: Problem 8

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=2}^{\infty} \frac{\ln \left(n^{2}\right)}{n} $$

Short Answer

Expert verified
The series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \) diverges.

Step by step solution

01

Verify Conditions for Integral Test

The Integral Test can be applied if the function \( f(x) = \frac{\ln(x^2)}{x} \) is positive, continuous, and decreasing for \( x \geq 2 \). \( f(x) \) is positive for \( x \geq 2 \) because both \( \ln(x^2) \) and \( x \) are positive for this range. \( \ln(x^2) \) is continuous as the composition of continuous functions, and \( x \) is continuous. To check if it鈥檚 decreasing, take the derivative of \( f(x) \).
02

Compute the Derivative

Calculate \( f'(x) \) where \( f(x) = \frac{\ln(x^2)}{x} \). Using the quotient rule, \( f'(x) = \frac{d}{dx} \left( \frac{\ln(x^2)}{x} \right) = \frac{x \cdot \frac{d}{dx}(\ln(x^2)) - \ln(x^2) \cdot \frac{d}{dx}(x)}{x^2} = \frac{x \cdot \frac{2}{x} - \ln(x^2)}{x^2} = \frac{2 - \ln(x^2)}{x^2} \). For \( x \geq 2 \), \( 2 - \ln(x^2) \leq 0 \), indicating \( f(x) \) is decreasing.
03

Set Up the Integral

Since the conditions are satisfied, apply the Integral Test. Evaluate the integral \( \int_{2}^{\infty} \frac{\ln(x^2)}{x} \, dx \), which simplifies to \( \int_{2}^{\infty} \frac{2\ln(x)}{x} \, dx \). This can be rewritten as \( 2 \int_{2}^{\infty} \frac{\ln(x)}{x} \, dx \).
04

Evaluate the Integral

Use substitution: let \( u = \ln(x) \), then \( du = \frac{1}{x}dx \). The integral becomes \( 2 \int_{\ln(2)}^{\infty} u \, du \), which evaluates to \( 2 [\frac{u^2}{2}]_{\ln(2)}^{\infty} = [u^2]_{\ln(2)}^{\infty} \).
05

Determine Divergence or Convergence

Evaluate \( [u^2]_{\ln(2)}^{\infty} = \infty - (\ln(2))^2 \), which diverges because the upper limit is infinite. Since the improper integral diverges, the series \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \) also diverges by the Integral Test.

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Key Concepts

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

Series Convergence
When dealing with series, we often need to determine whether they converge or diverge. Convergence means that as you add more and more terms, the total approaches a certain finite number. Divergence, on the other hand, implies that the total can keep increasing indefinitely without settling toward any particular value.
For a series like \( \sum_{n=2}^{\infty} \frac{\ln(n^2)}{n} \), using tests like the Integral Test helps us decide if the series converges or diverges. The Integral Test is useful because it allows us to compare a series to an improper integral. If the integral converges, so does the series; if the integral diverges, the series does too. This direct comparison simplifies the evaluation process considerably.
The series given in the exercise is analyzed by first ensuring that we can apply the Integral Test, which requires checking certain conditions for the related function is positive, continuous, and decreasing for the desired range.
Improper Integrals
Improper integrals extend the idea of definite integrals to functions with infinite limits or discontinuities in the range of integration. Evaluating an improper integral involves taking a limit to account for the infinity or discontinuity involved.
Taking the integral \( \int_{2}^{\infty} \frac{\ln(x^2)}{x} \, dx \) from the original exercise, we reformulate it using properties of logarithms and substitutions to find its exact behavior.
- If an integral leads to a finite number, it is considered convergent.
- If it results in an infinite value, it's divergent.
For the given integral, it diverges because as \( x \to \infty \), the logarithmic term increases without bound. Such divergence implies the series we are analyzing does not settle toward a finite value.
Calculus Derivatives
Understanding derivatives is crucial in many calculus concepts, including when determining if a function is decreasing. In the context of the Integral Test, we need to ensure the associated function does not increase for the test to be valid.
The derivation of \( f(x) = \frac{\ln(x^2)}{x} \) using the quotient rule is pivotal. The quotient rule tells us that if we have a function \( \frac{u(x)}{v(x)} \), the derivative is given by:\[ f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{v(x)^2} \]Applying it to our function, we're checking that the function is indeed decreasing for the range \( x \geq 2 \).
- A negative derivative implies that the function is decreasing.
The result \( f'(x) = \frac{2 - \ln(x^2)}{x^2} \) shows that this derivative is negative for \( x \geq 2 \), confirming the function is decreasing. This fact is essential for applying the Integral Test validly.

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Most popular questions from this chapter

By multiplying the Taylor series for \(e^{x}\) and \(\sin x,\) find the terms through \(x^{5}\) of the Taylor series for \(e^{x} \sin x .\) This series is the imaginary part of the series for \begin{equation} e^{x} \cdot e^{i x}=e^{(1+i) x} \end{equation} Use this fact to check your answer. For what values of \(x\) should the series for \(e^{x}\) sin \(x\) converge?

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \sin \frac{1}{n}\end{equation}

Suppose that \(a_{n}>0\) and \(\lim _{n \rightarrow \infty} a_{n}=\infty .\) Prove that \(\sum a_{n}\) diverges.

The limit \(L\) of an alternating series that satisfies the conditions of Theorem 15 lies between the values of any two consecutive partial sums. This suggests using the average $$\frac{s_{n}+s_{n+1}}{2}=s_{n}+\frac{1}{2}(-1)^{n+2} a_{n+1}$$ to estimate \(L .\) Compute $$s_{20}+\frac{1}{2} \cdot \frac{1}{21}$$ as an approximation to the sum of the alternating harmonic series. The exact sum is \(\ln 2=0.69314718 \ldots .\)

It is not yet known whether the series \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n}\end{equation} converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. \begin{equation} \begin{array}{l}{\text { a. Define the sequence of partial sums }}\end{array} \end{equation} \begin{equation} s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n}. \end{equation} \begin{equation} \begin{array}{l}{\text { What happens when you try to find the limit of } s_{k} \text { as } k \rightarrow \infty ?} \\ {\text { Does your CAS find a closed form answer for this limit? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Plot the first } 100 \text { points }\left(k, s_{k}\right) \text { for the sequence of partial }} \\ \quad {\text { sums. Do they appear to converge? What would you estimate }} \\ \quad {\text { the limit to be? }} \\ {\text { c. Next plot the first } 200 \text { points }\left(k, s_{k}\right) . \text { Discuss the behavior in }} \\ \quad {\text { your own words. }} \\ {\text { d. Plot the first } 400 \text { points }\left(k, s_{k}\right) \text { . What happens when } k=355 \text { ? }} \\\ \quad {\text { Calculate the number } 355 / 113 . \text { Explain from you calculation }} \\ \quad {\text { what happened at } k=355 . \text { For what values of } k \text { would you }} \\ \quad {\text { guess this behavior might occur again? }} \end{array} \end{equation}
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