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Chapter 7: Problem 64

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \ln \frac{1}{n} $$

Short Answer

Expert verified
The series diverges according to the test for divergence since the sequence of terms \( \ln \left(\frac{1}{n}\right) \) does not approach zero as \( n \) goes to infinity.

Step by step solution

01

Identify the sequence of terms

The first thing to do is to identify the sequence of terms in the series which is \( a_n = \ln \left(\frac{1}{n}\right) \) for \( n \geq 1 \)
02

Apply the test for divergence

Next, determine whether the sequence of terms approaches zero as \( n \) approaches infinity, that is \( \lim_{n\to\infty} a_n =? \). Evaluate the limit using the properties of logarithms \( \lim_{n\to\infty} \ln \left(\frac{1}{n}\right) = \ln \left(\lim_{n\to\infty} \frac{1}{n}\right) \). Since \( \frac{1}{n} \) approaches zero as \( n \) goes to infinity, the limit is \( \ln(0) \). However, the natural logarithm of zero is undefined.
03

Determine convergence or divergence

Since the sequence of terms does not approach zero, the test for divergence indicates that the series diverges.

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Key Concepts

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

Divergence Test
The Divergence Test is a simple and essential tool in determining whether an infinite series diverges. For a series \( \sum a_n \) to converge, it is necessary for the sequence of terms \( a_n \) to approach zero as the index \( n \) approaches infinity. If the sequence does not tend to zero, the series must diverge. This is often the first test applied when assessing the convergence of a series.

To use the divergence test:
  • Identify the sequence of terms \( a_n \) of the series.
  • Compute \( \lim_{n\to\infty} a_n \)
  • If \( \lim_{n\to\infty} a_n eq 0 \), the series diverges.
  • If \( \lim_{n\to\infty} a_n = 0 \), the test is inconclusive.
In our exercise, the terms are \( a_n = \ln \left(\frac{1}{n}\right) \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \), and since \( \ln(0) \) is undefined, the terms do not approach zero. Therefore, the series diverges.
Natural Logarithms
Natural logarithms (ln) are logarithms to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. The function \( \ln(x) \) describes the power to which \( e \) must be raised to obtain the number \( x \). Natural logarithms are widely applicable in mathematical expressions, particularly in calculus and analysis.

Some key properties of natural logarithms include:
  • The domain is \( (0, \infty) \), meaning \( x \) must be positive.
  • \( \ln(1) = 0 \), since \( e^0 = 1 \).
  • \( \ln(ab) = \ln(a) + \ln(b) \) - the product rule.
  • \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) - the quotient rule.
  • \( \ln(a^b) = b \ln(a) \) - the power rule.
In our exercise, the term \( \ln\left(\frac{1}{n}\right) \) involves the logarithmic function applied to a fraction. As \( n \) increases, the argument of the logarithm \( \frac{1}{n} \) tends towards zero, highlighting the properties and behavior of natural logs.
Infinite Series
An infinite series is a sum of an infinite sequence of terms \( a_1, a_2, a_3, \ldots \) continuing indefinitely. Infinite series are a fundamental concept in mathematics, especially in calculus and analysis, used to describe many natural and theoretical phenomena.

There are different types of series:
  • Convergent series: The sum approaches a specific number as more terms are added.
  • Divergent series: The sum either increases without bound or oscillates without approaching any value.
  • Conditionally convergent series: Converges only under particular arrangements of terms.
To determine the behavior of a series:
  • Apply tests like the divergence test, which checks if terms go to zero.
  • Sequence of partial sums, which helps visualize convergence or divergence.
  • Techniques such as the ratio test, root test, or comparison test for more complex assessment.
In our scenario, the series \( \sum_{n=1}^{\infty} \ln \frac{1}{n} \) diverges as its sequence of terms does not fulfill the requirement of approaching zero.

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Most popular questions from this chapter

Modeling Data The annual sales \(a_{n}\) (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) represents the year, with \(n=3\) corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report) (3,3844),(4,4267),(5,4492),(6,4814),(7,5079) (8,5213),(9,5289),(10,5682),(11,5958),(12,6171) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=3,4, \ldots, 12\) for the data. Graphically compare the points and the model. (b) Use the model to predict sales in the year 2008 .

Prove, using the definition of the limit of a sequence, that \(\lim _{n \rightarrow \infty} r^{n}=0\) for \(-1

Use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ You go to work at a company that pays \(\$ 0.01\) for the first day, \(\$ 0.02\) for the second day, \(\$ 0.04\) for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\) (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by \(b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1\) (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho .\) Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Compound Interest Consider the sequence \(\left\\{A_{n}\right\\}\) whose \(n\) th term is given by \(A_{n}=P\left(1+\frac{r}{12}\right)^{n}\) where \(P\) is the principal, \(A_{n}\) is the account balance after \(n\) months, and \(r\) is the interest rate compounded annually. (a) Is \(\left\\{A_{n}\right\\}\) a convergent sequence? Explain. (b) Find the first 10 terms of the sequence if \(P=\$ 9000\) and \(r=0.055\)
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