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Logarithmic differentiation is a clever method that helps simplify the differentiation of complex functions, particularly those involving products or powers of variables.
It is based on taking the natural logarithm of a function, which turns products into sums and exponents into multiplication, making differentiation more straightforward.

For example, if you have a function like \(f(n) = \left(1 + \frac{1}{n}\right)^n\), directly differentiating it can be tricky.
By logarithmically differentiating, you first take the natural log:

• \(\ln f(n) = n \ln \left(1 + \frac{1}{n}\right)\)

This transformation makes it easier to apply differentiation rules like the product and chain rules.
Such an approach is vital when working with sequences such as \(x_n = \left(1 + \frac{1}{n}\right)^n\), because it simplifies the steps needed to understand the behavior of the sequence.
In this context, if \(\ln f(n)\) is increasing, then \(f(n)\) itself is increasing, which is a crucial insight when proving properties like monotonicity.

Sequence convergence refers to the concept of whether a sequence approaches a specific value as its index grows larger.
A sequence is termed "convergent" if it approaches a particular limit.
In mathematical terms, for a sequence \( \{a_n\} \), it converges to a limit \( L \) if for every positive number \( \epsilon \), there exists an integer \( N \) such that for all \( n > N \), the terms of the sequence satisfy \( |a_n - L| < \epsilon \).

In the case of the sequence \( x_n = \left(1 + \frac{1}{n}\right)^n\), it is known to converge to the mathematical constant \( e \) (approximately 2.718).
This conclusion comes from both theoretical work related to the definition of \( e \) and practical numerical computations showing \( x_n \) getting increasingly closer to \( e \) as \( n \) increases.
The convergence is significant because it connects this seemingly simple sequence to one of the most important constants in mathematics.

Derivative analysis is a technique used to understand how a function behaves as its input changes — essentially, it reveals the rate of change.
Calculating the derivative provides vital insights into whether a function is increasing or decreasing.
A positive derivative indicates an increasing function, while a negative derivative points to a decreasing one.

In this exercise, we applied derivative analysis to the function \( g(n) = n \ln \left(1 + \frac{1}{n}\right) \) to examine if it is monotone increasing.
The key step here was to find \( g'(n) \), the derivative of \(g(n)\) with respect to \( n\), which we simplified as:

• \(g'(n) = \ln\left(1 + \frac{1}{n}\right) - \frac{1}{n+1}\)

By assessing the derivative, we confirmed it holds positive values for all larger values of \( n \), establishing that the sequence \( x_n = \left(1 + \frac{1}{n}\right)^n \) is indeed monotone increasing.
This analysis illuminates the behavior of sequences and functions, offering a deeper comprehension beyond numerical computation alone.