91Ó°ÊÓ

When analyzing sequences, one important concept to understand is the **limit of a sequence**. The limit of a sequence is a value that the elements of the sequence approach as the index goes to infinity. In simpler terms, it is where the sequence "settles down" or becomes stable.
If such a value exists, the sequence is said to converge to that limit. Otherwise, the sequence diverges. In the exercise, the sequence is defined as \(a_n = \ln \left( \frac{n}{n+1} \right)\).
To determine convergence, we analyzed what happens to the expression \(\frac{n}{n+1}\) as \(n\) grows larger. Simplifying it, we found that \(\frac{n}{n+1}\) approaches \(1\) when \(n\) approaches infinity. Consequently, the sequence converges to the limit \(\ln(1)\), which is \(0\).

Logarithms are powerful mathematical tools that help us manipulate exponential expressions. One useful property is the difference of logs: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).
This property allows us to simplify expressions involving logarithms, as seen in the exercise. Originally, \(a_n = \ln n - \ln (n+1)\).
Using the log property, this expression became \(\ln \left( \frac{n}{n+1} \right)\). This reduced form is much easier to analyze when finding convergence.

• It saves time by simplifying calculations.
• Makes the problem more tractable.
• Allows for easier limits evaluation.

Mastering these properties can greatly benefit students in solving a wide range of mathematical problems.

Infinite limits relate to the behavior of a sequence or function as it approaches infinity. It’s crucial in understanding whether sequences or functions grow without bound or stabilize at a specific value.
In the given problem, we needed to find the behavior of \(\frac{n}{n+1}\) as \(n\) becomes large.
Rewriting this expression as \(\frac{1}{1+\frac{1}{n}}\) reveals its behavior as \(n\to\infty\):

• \(\frac{1}{n} \to 0\), thus \(1 + \frac{1}{n} \to 1\).
• The entire fraction \(\frac{n}{n+1} \to 1\).

The logarithm of \(1\) is \(0\), indicating the sequence converges. Understanding infinite limits is essential because it often reveals hidden patterns and outcomes of sequences and functions as they stretch infinitely.