/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Determine whether the series con... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 19

Determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{\tan ^{-1} k}{1+k^{2}} $$

Short Answer

Expert verified
The series converges by the Comparison Test.

Step by step solution

01

Identify the Series

We are given the series \( \sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1+k^{2}} \). Our goal is to determine whether this series converges.
02

Apply the Comparison Test

Notice that for large \( k \), \( \tan^{-1} k \) approaches \( \frac{\pi}{2} \). Therefore, each term \( a_k = \frac{\tan^{-1} k}{1+k^{2}} \) behaves roughly like \( \frac{\frac{\pi}{2}}{k^2} \). Consider the series \( \sum_{k=1}^{\infty} \frac{\frac{\pi}{2}}{k^2} \), which is a constant multiple of the convergent p-series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) with \( p = 2 > 1 \).
03

Comparing Terms

For sufficiently large \( k \), \( \frac{\tan^{-1} k}{1+k^{2}} \leq \frac{\frac{\pi}{2}}{k^2} \). Since \( \sum_{k=1}^{\infty} \frac{\frac{\pi}{2}}{k^2} \) converges, by the Comparison Test, \( \sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1+k^{2}} \) also converges.
04

Conclusion

Since we have established that \( \sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1+k^{2}} \leq \sum_{k=1}^{\infty} \frac{\frac{\pi}{2}}{k^2} \), and since the latter series is known to converge, we conclude that the given series converges by the Comparison Test.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Comparison Test
The Comparison Test is a powerful tool used to determine the convergence or divergence of a series by comparing it to another series with known behavior. When using the Comparison Test, we typically look at two non-negative series, say \( \sum a_k \) and \( \sum b_k \). To apply:
  • We first need each term of one series to be less than or equal to the corresponding term in the series with known behavior for all sufficiently large index values.
  • If \( a_k \leq b_k \) and \( \sum b_k \) converges, then \( \sum a_k \) also converges.
  • Conversely, if \( a_k \geq b_k \) and \( \sum b_k \) diverges, then \( \sum a_k \) also diverges.

In our problem, we compared \( \sum \frac{\tan^{-1} k}{1+k^{2}} \) to the known convergent series \( \sum \frac{\frac{\pi}{2}}{k^2} \) because it demonstrated a similar behavior for large \( k \). Hence, using the Comparison Test, we concluded that our series converges.
p-series
A p-series is a type of infinite series that takes the form \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) where \( p\) is a real number. The convergence of p-series depends largely on the value of \( p \):
  • If \( p > 1\), the series converges.
  • If \( p \leq 1\), the series diverges.
In the given exercise, we considered the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 \). Since \( 2 > 1 \), it is known to converge.
Recognizing this, we used it as a benchmark (through the Comparison Test) to assess the convergence of our original series, \( \sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1+k^{2}} \). This approach simplifies the evaluation of the series' behavior, ensuring students understand its application in convergence tests.
Arctan Function
The Arctan function, or \( \tan^{-1} x \), is the inverse of the tangent function. Its range is limited, spanning from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). In mathematical series, the behavior of the Arctan function for large values of \( x \) is particularly relevant:
  • As \( x \) grows larger, \( \tan^{-1} x \) approaches \( \frac{\pi}{2} \).
  • This limit leads the fraction \( \frac{\tan^{-1} k}{1+k^{2}} \) to behave like \( \frac{\frac{\pi}{2}}{k^2} \), particularly as \( k \) becomes very large.
Understanding the asymptotic nature of the Arctan function helps in approximating the terms of a series, which is crucial when applying convergence tests such as the Comparison Test. By recognizing this, we accurately compared and estimated the convergence of given series, using known convergent patterns like those found in a p-series.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the domain of the function $$ f(x)=\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{(2 k-2) !} x^{k} $$

(a) Use the relationship $$ \int \frac{1}{\sqrt{1+x^{2}}} d x=\sinh ^{-1} x+C $$ to find the first four nonzero terms in the Maclaurin series for \(\sinh ^{-1} x\) (b) Express the series in sigma notation. (c) What is the radius of convergence?

Research has shown that the proportion \(p\) of the population with IQs (intelligence quotients) between \(\alpha\) and \(\beta\) is approximately $$ p=\frac{1}{16 \sqrt{2 \pi}} \int_{\alpha}^{\beta} e^{-\frac{1}{2}\left(\frac{x-100}{16}\right)^{2}} d x $$ Use the first three terms of an appropriate Maclaurin series to estimate the proportion of the population that has \(\mathrm{IQs}\) between 100 and 110 .

Confirm the integration formula by integrating the appropriate Maclaurin series term by term. (a) \(\int \sin x d x=-\cos x+C\) (b) \(\int \frac{1}{1+x} d x=\ln (1+x)+C\)

Find the first four nonzero terms of the Maclaurin series for the function by making an appropriate substitution in a known Maclaurin series and performing any algebraic operations that are required. State the radius of convergence of the series. (a) \(\frac{x^{2}}{1+3 x}\) (b) \(x \sinh 2 x\) (c) \(x\left(1-x^{2}\right)^{3 / 2}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.