Question

Using the binomial theorem, find the limit of Sn=(1+1/n)^n-1

Answer 1

$$\lim_{n \to \infty} \left[ \left(1 + \frac{1}{n}\right)^n - 1 \right] = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n - 1 = \left[1^n \right] - 1 = e^{\lim_{n \to \infty} \frac{n \left(1 + \frac{1}{n} - 1\right)}{n}} - 1 = e^{\lim_{n \to \infty} \frac{n \left( \frac{1}{n} \right)}{n}} - 1 = e - 1.$$

Answer: $e - 1$.