1. The problem is to understand how the mathematical constant $e$ is derived. 2. The number $e$ is defined as the limit of $\left(1 + \frac{1}{n}\right)^n$ as $n$ approaches infinity. 3. This can be written as: $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ 4. To understand this, consider the expression for increasing values of $n$: - For $n=1$, $\left(1 + \frac{1}{1}\right)^1 = 2$ - For $n=10$, $\left(1 + \frac{1}{10}\right)^{10} \approx 2.5937$ - For $n=100$, $\left(1 + \frac{1}{100}\right)^{100} \approx 2.7048$ - For $n=1000$, $\left(1 + \frac{1}{1000}\right)^{1000} \approx 2.7169$ 5. As $n$ becomes very large, the value approaches approximately 2.71828, which is the constant $e$. 6. Another way to define $e$ is through the infinite series: $$e = \sum_{k=0}^\infty \frac{1}{k!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$ 7. This series converges to the same value of $e$. 8. The constant $e$ is fundamental in calculus, especially in continuous growth and decay problems, and is the base of natural logarithms. Final answer: $e \approx 2.71828$