91Ó°ÊÓ

To determine if a sequence converges, we often have to evaluate limits. Sometimes, we encounter indeterminate forms like \(\frac{\text{ln}(n)}{n^2} \) which can be tricky.

This is where L'Hôpital's Rule is incredibly useful. L'Hôpital's Rule allows us to handle limits of forms \(\frac{0}{0} \) or \(\frac{\text{∞}}{\text{∞}} \). In simple terms, if a limit is hard to evaluate directly, you can differentiate the numerator and the denominator separately and then take the limit again.

For example, let's look at our sequence's limit: \(\text{lim}_{n \to \theta} \frac{\text{ln}(n)}{n^2} \). Here, both the numerator ln(n) and the denominator n² approach ∞, giving an \(\frac{\text{∞}}{\text{∞}} \) form. Therefore, we can use L'Hôpital's Rule. Differentiating ln(n) gives \(\frac{1}{n} \), and differentiating n² gives 2n, leading us to the new limit: \(\text {lim}_{n \to ∞} \frac{1/n}{2n} = \text {lim}_{n \to ∞} \frac{1}{2n^2} \).

Evaluating limits is a key part of understanding sequence convergence. Here's a step-by-step approach to evaluate limits, particularly for our sequence \(\frac{\text{ln}(n)}{n^2} \):

1. **Identify the form**: The given sequence was in an \(\frac{\text{∞}}{\text{∞}} \) form, recognizable because both the logarithmic and quadratic functions grow very large as n increases.

2. **Apply appropriate rule**: Since we have an indeterminate form, we used L'Hôpital's Rule. We took derivatives of the numerator and the denominator separately.

3. **Reevaluate the limit**: After applying L'Hôpital's Rule, we simplified the new expression to \(\frac{1}{2n^2} \) and then reevaluated the limit as n approaches ∞.

4. **Determine convergence**: Once simplified, it was clear that as the denominator grows (n^2 gets very large), the entire fraction approaches 0.

A sequence is said to be convergent if it approaches a specific value as n becomes very large. In our case,
we've concluded that \(\frac{\text{ln}(n)}{n^2} \) converges to 0. But why?

1. **Behavior of Functions**: Logarithmic functions, like ln(n), grow slower compared to polynomial functions like n². As a result, the denominator grows much faster than the numerator.

2. **Limit Explanation**: As n increases, \(\frac{1}{2n^2} \) becomes smaller and smaller. Hence, it approaches 0. This is fundamentally why the sequence converges to 0.

3. **Real-Life Implications**: Understanding convergent sequences helps in fields like physics, engineering, and even finance, where knowing the long-term behavior of sequences is crucial.