Question

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{20} n}\right\\}$$

Short answer

Answer: The sequence diverges, as the limit is infinity.

Step-by-step solution

Determining if the sequence is in an indeterminate form

As the sequence is given by: $$\left\\{\frac{n^{10}}{\ln ^{20} n}\right\\}$$ We can clearly see that both the numerator and denominator go to infinity as \(n\) approaches infinity. Therefore, the sequence is in an indeterminate form, and we can proceed by applying L'Hopital's rule.

Applying L'Hopital's Rule

First, let's differentiate both the numerator and the denominator with respect to \(n\). We get: $$\frac{d(n^{10})}{dn} = 10n^9$$ $$\frac{d(\ln^{20} n)}{dn} = \frac{20\ln^{19} n}{n}$$ Now, let's compute the limit of the ratio of the derivatives as \(n\) approaches infinity: $$\lim_{n \to \infty}\frac{10n^9}{\frac{20\ln^{19} n}{n}}$$

Simplifying the expression

To simplify the expression, let's multiply the numerator and the denominator by \(n\): $$\lim_{n \to \infty}\frac{10n^{10}}{20\ln^{19} n}$$ Now, we can simplify the expression further: $$\lim_{n \to \infty}\frac{n^{10}}{2\ln^{19} n}$$

Determining whether the limit converges or diverges

Now, let's analyze the limit. Notice that as \(n\) approaches infinity, the term \(n^{10}\) grows at a much faster rate compared to the term \(\ln^{19} n\). Therefore, the limit of the sequence as \(n\) approaches infinity is infinity: $$\lim_{n \to \infty}\frac{n^{10}}{2\ln^{19} n} = \infty$$ The sequence diverges, as the limit is infinity.

Key concepts

Sequences

In mathematics, a sequence is a list of numbers that follows a specific pattern or rule. A sequence can be thought of as a list of numbers in a particular order. Each number in the sequence is called a term. Sequences can be finite or infinite. An infinite sequence goes on forever, and is what we often deal with in calculus when we want to find the limit of a sequence.
For instance, the sequence given by \( \left\{ \frac{n^{10}}{\ln^{20} n} \right\} \) is infinite, depending on the variable \( n \), which increases without bounds. This particular sequence has terms that are fractions whose values change as \( n \) changes. Understanding sequences is crucial when discussing limits because what happens to the terms of a sequence as they "approach infinity" can help us understand the behavior of functions and other mathematical constructs over a long stretch of numbers.
As \( n \) becomes very large, the behavior of each term governs the behavior of the sequence as a whole. In calculus, we often seek to find whether a sequence converges to a finite number or diverges, in which case it "runs off" to infinity or fails to settle down to one specific value.

Indeterminate Forms

Indeterminate forms occur when evaluating a limit results in expressions like \( \frac{\infty}{\infty} \), \( 0 \times \infty \), or \( \infty - \infty \). These forms are considered indeterminate because they do not immediately reveal information about the limit's value without further analysis or manipulation.
The sequence we are examining, \( \left\{ \frac{n^{10}}{\ln^{20} n} \right\} \), as \( n \to \infty \), gives rise to an indeterminate form because both the numerator \( n^{10} \) and the denominator \( \ln^{20} n \) become very large. In mathematics, particularly calculus, certain methods like L'Hopital's Rule can be employed to resolve these indeterminate forms. This rule allows us to differentiate the numerator and the denominator separately and then evaluate the limit as \( n \to \infty \).
Indeterminate forms require such techniques because the surface level computation doesn’t clearly show whether a limit exists or not. Thus, we perform additional steps like differentiating to get a more straightforward expression that is easier to evaluate.

Limits

In calculus, a limit is a concept that describes the behavior of a function or sequence as its input gets closer to a particular point or approaches infinity. Limits are foundational to understanding calculus as they form the basis for the definition of derivatives and integrals.
Let's consider the limit of the sequence \( \left\{ \frac{n^{10}}{\ln^{20} n} \right\} \). By using L'Hopital's Rule, we determine the limit of this sequence as \( n \to \infty \). After simplifying the expression using L'Hopital's Rule, we're left with \( \lim_{n \to \infty}\frac{n^{10}}{2\ln^{19} n} \). By examining the growth rates, we conclude that \( n^{10} \) outpaces \( \ln^{19} n \), implying that the limit approaches infinity.
This conclusion about the limit tells us that the sequence diverges. Limits help identify whether sequences or functions approach a specific value or grow unbounded. Understanding limits provides insight into the long-term behavior of sequences and functions, which is crucial for higher-level mathematics and practical applications alike.