Question

Use Theorem 6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{1000} n}\right\\}$$

Short answer

Question: Determine the limit of the sequence $$\left\\{\frac{n^{10}}{\ln ^{1000} n}\right\\}$$ as $$n$$ approaches infinity, or state if it diverges. Answer: The limit of the sequence $$\left\\{\frac{n^{10}}{\ln ^{1000} n}\right\\}$$ as $$n$$ approaches infinity converges to 0.

Step-by-step solution

Define sequences and the given limits

Let us define the sequences $$\{a_n\} = \{n^{10}\}$$ and $$\{b_n\} = \{\ln ^{1000} n\}$$. Now, we have to find the limits of the ratios $$\frac{a_{n+1}}{a_n}$$ and $$\frac{b_{n+1}}{b_n}$$ as $$n$$ approaches infinity.

Compute the limits of the ratios of the sequences

We first calculate the limit of the ratio $$\frac{a_{n+1}}{a_n}$$: $$\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{(n+1)^{10}}{n^{10}} = \lim_{n\to\infty} \left(1+ \frac{1}{n}\right)^{10} = 1^{10} = 1$$ Next, we need to find the limit of the ratio $$\frac{b_{n+1}}{b_n}$$: $$\lim_{n\to\infty} \frac{b_{n+1}}{b_n} = \lim_{n\to\infty} \frac{\ln^{1000}(n+1)}{\ln^{1000}n} = \lim_{n\to\infty} \left(\frac{\ln(n+1)}{\ln n}\right)^{1000}$$

Applying L'Hôpital's rule to solve limit

This limit is in indeterminate form, so we will use L'Hôpital's rule by taking the derivative of the numerator and denominator with respect to n: $$\lim_{n\to\infty} \frac{1000 \frac{1}{n+1} \left(\frac{\ln(n+1)}{\ln n}\right)^{999}}{1000 \frac{1}{n} \left(\frac{\ln(n+1)}{\ln n}\right)^{999}}$$ Now, cancel out 1000 and $$(\frac{\ln(n+1)}{\ln n})^{999}$$: $$\lim_{n\to\infty} \frac{(n+1)}{n} = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right) = 1$$

Compare the limits and determine convergence or divergence

Since $$\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 1$$ and $$\lim_{n\to\infty} \frac{b_{n+1}}{b_n} = 1$$, we cannot conclude that $$\lim_{n\to\infty} \frac{n^{10}}{\ln ^{1000} n} = 0$$ using Theorem 6. In this case, we have to determine the convergence or divergence using another method. We can apply L'Hôpital's rule multiple times to our original sequence: $$\lim_{n\to\infty} \frac{n^{10}}{\ln ^{1000} n}$$ After applying L'Hôpital's rule 10 times, we get: $$\lim_{n\to\infty} \frac{10!}{(\frac{1000}{n^2})^{10} \ln^{990} n}$$ As $$n$$ approaches infinity, the denominator becomes infinitely large, while the numerator remains constant. Therefore, the limit of the sequence converges to 0: $$\lim_{n\to\infty} \frac{n^{10}}{\ln ^{1000} n}=0$$

Key concepts

Limits of Sequences

Understanding the limits of sequences is crucial when studying calculus and numerical series. A sequence is essentially an ordered list of numbers, where each number is called a term. As we progress through this list, identifying the behavior of terms as they approach infinity is key. This behavior is captured by the sequence's limit. In mathematical terms, if a sequence \( \{a_n\} \) has a limit \( L \), it means that as \( n \) becomes very large, the terms \( a_n \) get arbitrarily close to \( L \).

In some cases, sequences converge, meaning they approach a specific limit. Alternatively, a sequence diverges if it doesn't settle towards a particular value. Calculating these limits often involves breaking complex expressions into simpler pieces, allowing us to understand how they behave as \( n \) grows.

• If the terms get closer to a specific number, the sequence converges.
• If the terms grow without bound or oscillate, the sequence diverges.

Having a systematic method to explore limits, like employing limit laws and rules, makes the process more accessible and understandable.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used to solve limits that result in indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter these forms, direct substitution in a limit leads to ambiguity. That's where L'Hôpital's Rule comes into play. This rule states that if the limits of the numerator and denominator both approach \( 0 \) or \( \infty \), the limit of the quotient can be found by taking the derivatives.

To apply L'Hôpital's Rule:

• First, ensure both the numerator and denominator approach \( 0 \) or \( \infty \).
• Take the derivative of the numerator and the derivative of the denominator independently.
• Calculate the limit of the new ratio.

L'Hôpital's Rule can often be applied multiple times to resolve especially complicated expressions. For example, when analyzing the limit of \( \frac{n^{10}}{\ln^{1000} n} \), repeated application of L'Hôpital's Rule allows us to methodically simplify until an evaluable limit is achieved, ultimately leading to convergence towards zero. This method streamlines complex evaluations and removes ambiguity from indeterminate forms.

Convergence and Divergence

Convergence and divergence are concepts central to evaluating sequences and series. A sequence is said to converge if its terms approach a specific number as \( n \) becomes very large. This number is known as the limit of the sequence. In contrast, divergence occurs when no such limit exists, meaning the terms either increase or decrease without bound, or oscillate indefinitely without settling on a value.

To determine convergence or divergence, several tests and rules can be applied:

• Comparison Test: Compares the given sequence with another known convergent or divergent sequence.
• Ratio Test: Analyzes the ratio of consecutive terms.
• Integral Test: Uses integrals to assess convergence.

In the given exercise, applying repeated differentiation through L'Hôpital's Rule helped identify that the sequence \( \frac{n^{10}}{\ln^{1000} n} \) ultimately converges to 0. Hence, even though directly observing the sequence might not immediately indicate convergence, using mathematical operations like L'Hôpital’s Rule clarified the outcome, revealing the sequence's behavior over a long span. Understanding these concepts helps demystify the behavior of sequences and powerfully predicts their tendencies as they stretch to infinity.