limits

Motivation My previous post about the definition of the exponential function has provided no connection between a well-known characterization (or an alternative definition) of the exponential function: $$\exp(s) = \lim_{n\to\infty} \sum_{k=1}^n \frac{s^k}{k!}.$$ The term to be summed is simpler than the one in the binomial expansion of $(1 + s / n)^n$. Solution We want this sum to be as small as possible as $n \to \infty$. $$\sum_{k=2}^n \left( 1 - \frac{n \cdot \dots \cdot (n + 1 - k)} {\underbrace{n \cdot \dots \cdot n}_{n^k}} \right) \, \frac{x^k}{k! [Read More]

Motivation from compound interest Increase the number of times that the interest is compounded each year ($n$), so as to increase the final amount of money ($A$) $$A = P \left(1 + \frac{r}{n}\right)^{nt}.$$ $t$ and $r$ are the number of years and the interest rate per annum. As $n$ becomes large, we can approximate the amount by $$A = P e^{rt}.$$ The value of $A$ in the above formula is the amount compounded continuously. [Read More]