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Chapter 8: Problem 56

Determine whether the series converges absolutely, converges conditionally, or diverges. The tests that have been developed in Section 5 are not the most appropriate for some of these series. You may use any test that has been discussed in this chapter. \(\sum_{n=1}^{\infty}(-1)^{n} \ln (2+1 / n)\)

Short Answer

Expert verified
The series diverges because the terms do not go to zero.

Step by step solution

01

Identify the series

The given series is \(\sum_{n=1}^{\infty} (-1)^{n} \ln \left(2 + \frac{1}{n}\right)\). This is an alternating series because of the term \((-1)^n\).
02

Apply the Alternating Series Test

The Alternating Series Test states that an alternating series \(\sum (-1)^n a_n\) converges if the absolute values \(a_n\) decrease monotonically to 0. Here, \(a_n = \ln \left(2 + \frac{1}{n}\right)\).
03

Analyze \(a_n\) behavior as \(n\to\infty\)

As \(n\) increases, \(\frac{1}{n}\) gets smaller, so \(2 + \frac{1}{n}\) approaches 2. Thus, \(\ln\left(2 + \frac{1}{n}\right)\) approaches \(\ln(2)\), which is not zero. Therefore, \(a_n\) does not go to 0.
04

Conclusion by the Alternating Series Test

Since \(a_n\) does not converge to 0, the Alternating Series Test fails. Therefore, the series \(\sum_{n=1}^{\infty} (-1)^{n} \ln \left(2 + \frac{1}{n}\right)\) diverges.

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Key Concepts

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

Convergence Tests
Convergence tests are essential tools in calculus to determine whether a sequence or series converges or diverges. These tests help analyze the behavior of series, providing insights into their ultimate nature. Some popular convergence tests include:
  • Ratio Test: This test evaluates the limit of the ratio of successive terms. If the limit is less than 1, the series converges absolutely.
  • Root Test: This examines the nth root of the absolute value of the nth term. Similar to the Ratio Test, if the limit is less than 1, the series converges absolutely.
  • Alternating Series Test: Used for series with terms alternating in sign, it checks if the absolute value terms decrease and approach zero as n increases.
Learning how to use convergence tests effectively enhances one's ability to work with infinite series. Using the correct test means looking at the series鈥� structure and evaluating the terms carefully. It requires both practice and understanding of the series鈥� characteristics.
Divergence
Divergence is the opposite of convergence, indicating that a series does not settle into a fixed value or pattern. A series diverges when the sum either increases without bound, decreases without bound, or oscillates without approaching a particular limit.
To identify divergence, we can consider:
  • Term Test for Divergence: If the nth term of a series does not approach zero as n approaches infinity, the series diverges.
  • Checking individual term behavior can also highlight if a series diverges.
In the case of the series \(\sum_{n=1}^{\infty}(-1)^{n} \ln \left(2+\frac{1}{n}\right)\), the ln function approaches a non-zero constant value, leading to divergence. Properly identifying divergence is key to understanding series that do not converge.
Alternating Series Test
The Alternating Series Test is a convergence test applicable to series with terms that change signs between positive and negative. A series of the form \(\sum (-1)^n a_n\) can be checked with this test by following two conditions:
  • The absolute values \[ a_n \]\ of the series terms should decrease monotonically as \(n\rightarrow\infty\).
  • The limit of \[ a_n \]\ as \(n\rightarrow\infty\) should be zero.
If both conditions are met, the series converges. In our original example \(\sum_{n=1}^{\infty} (-1)^{n} \ln \left(2 + \frac{1}{n}\right)\), the series failed the test because the terms settle at a non-zero value. Knowing how to apply the Alternating Series Test provides a clear pathway to determine convergence in many series where sign alternation is present.
Series Convergence
Series convergence involves the sum of infinite terms approaching a finite limit. A convergent series adds up to a specific value, contrasting with a divergent series, which does not. To determine series convergence, it is crucial to evaluate the terms and use appropriate convergence tests.
Convergence is categorized into two main types:
  • Absolute Convergence: A series converges absolutely if the series of absolute values converges. This implies that rearranging the terms does not affect the sum.
  • Conditional Convergence: A series converges conditionally if it converges, but not absolutely. This often occurs in alternating series.
Understanding the convergence of series enables mathematicians and students alike to work effectively with infinite sums and identify their ultimate behavior. This insight into series also underscores the connection between infinite works and finite solutions in mathematics.

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Most popular questions from this chapter

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x^{2}} $$

Use the Uniqueness Theorem to determine the coefficients \(\left\\{a_{n}\right\\}\) of the solution \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the given initial value problem. \(d y / d x=2 y \quad y(0)=3\)

In each of Exercises 43-48, write Newton's binomial series for the given expression up to and including the \(x^{3}\) term. $$ 1 /(1+x)^{2} $$

Suppose that every dollar that we spend gives rise (through wages, profits, etc.) to 90 cents for someone else to spend. That 90 cents will generate a further 81 cents for spending, and so on. How much spending will result from the purchase of a \(\$ 16,000\) automobile, the car included? (This phenomenon is known as the multiplier effect.)

In each of Exercises 91-94 a function \(f\), a base point \(c\), and a point \(x_{0}\) are given. Plot \(y=\left|f^{(3)}(t)\right|\) for \(t\) between \(c\) and \(x_{0}\). Use your plot to estimate the quantity \(M\) of Theorem 2 . Then use your value of \(M\) to obtain an upper bound for the absolute error \(\left|R_{2}\left(x_{0}\right)\right|=\left|f\left(x_{0}\right)-T_{2}\left(x_{0}\right)\right|\) that results when \(f\left(x_{0}\right)\) is approximated by the order 2 Taylor polynomial with base point \(c\). $$ f(x)=\sqrt{16+x^{2}} \quad c=3 \quad x_{0}=2.4 $$
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