Problem 89

Question

Which of the sequences $\left\\{a_{n}\right\\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{n} \int_{1}^{n} \frac{1}{x} d x $$

Step-by-Step Solution

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Answer

The sequence converges, and the limit is 0.

1Step 1: Understand the Integral Expression

The sequence is given as $ a_n = \frac{1}{n} \int_1^n \frac{1}{x} \, dx $. The integral $ \int_1^n \frac{1}{x} \, dx $ represents the natural logarithm function evaluated between 1 and $ n $, which simplifies to $ \ln(n) $.

2Step 2: Simplify the Sequence Expression

With the integral evaluated, we substitute $ \ln(n) $ back into the sequence expression, obtaining $ a_n = \frac{1}{n} \ln(n) $.

3Step 3: Analyze the Behavior as n Approaches Infinity

Consider the limit of $ a_n $ as $ n \to \infty $. We have \[ \lim_{{n \to \infty}} \frac{\ln(n)}{n} \], which is a standard indeterminate form $ \frac{\infty}{\infty} $. We can apply L'Hôpital's Rule to resolve the limit.

4Step 4: Apply L'Hôpital's Rule

Using L'Hôpital's Rule, we differentiate the numerator and the denominator: \[ \lim_{{n \to \infty}} \frac{\ln(n)}{n} = \lim_{{n \to \infty}} \frac{1/n}{1} = \lim_{{n \to \infty}} \frac{1}{n} = 0. \] Thus, the limit is 0.

5Step 5: Conclusion on Convergence

Since $ a_n \to 0 $ as $ n \to \infty $, the sequence $ a_n $ converges. The limit of the sequence is 0.

Key Concepts

Convergent SequencesLimits of SequencesL'Hôpital's Rule

Convergent Sequences

In the study of sequences in calculus, a sequence is considered convergent if it approaches a specific value as the number of terms goes to infinity. Essentially, as you progress further in the sequence, the terms get closer and closer to a particular number, which we call the limit of the sequence.
Understanding convergence is critical as it allows us to predict the behavior of sequences in various mathematical contexts.When a sequence converges, it does not have to reach a limit in a finite number of steps. Instead, it simply needs to approach it as closely as we desire by considering sufficiently many terms.
In our exercise, we analyzed the sequence defined by:

The sequence formula: \[ a_n = \frac{1}{n} \ln(n) \]

By examining the behavior of $a_n$ as $n$ becomes very large, we found that this sequence converges to 0 as explained through applying limits and L'Hôpital's Rule.

Limits of Sequences

Limits are fundamental in mathematics, especially when dealing with sequences and their long-term behavior. The limit of a sequence $\{a_n\}$ is defined as the value that the terms of the sequence approach as $n$ increases indefinitely. Mathematically, it is expressed as:

$ \lim_{{n \to \infty}} a_n = L $

where $L$ is the limit of the sequence. When a sequence has a limit, it is known to be convergent. For our exercise, we needed to calculate the limit of the sequence $a_n = \frac{1}{n}\ln(n)$. To do this, we looked at what happens to $\ln(n)$ and $1/n$ separately as $n$ goes to infinity.

$\ln(n)$ grows without bound (approaches infinity),
$\frac{1}{n}$ shrinks to zero.

Combining these, we got an indeterminate form $\frac{\infty}{\infty}$ when calculating $\lim_{{n \to \infty}} \frac{\ln(n)}{n}$. Using the techniques like L'Hôpital's Rule, we found the limit, which helped determine the behavior of the sequence.

L'Hôpital's Rule

L'Hôpital's Rule is a method in calculus for finding limits of indeterminate forms, those involving expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. This rule is particularly useful for sequences that do not yield a straightforward limit solution through simple substitution or algebraic manipulation. To apply L'Hôpital's Rule, you first need to differentiate the numerator and the denominator separately and then take the limit again.
In the exercise, we faced the limit