Algebra For Quant

I am storing here a few nuggets of algebra, I need for quantitative finance and machine learning.

\(e^x\) as an infinite serie

Using the McLaurin series expansion, we can define \(e^x\) as an infinite sum.
Here is how it goes: \[e^x \approx f(0) + f'(0) \frac{x}{1} + f''(0) \frac{x^2}{2!} + f'''(0) \frac{x^3}{3!} + \cdots + f^n(0) \frac{x^n}{n!}\]

As \(f(0) = f'(0) = f''(0) = f^n(0) = e^0 = 1\), we can rewrite our previous expression as

\[e^x \approx 1 + \frac{x}{1} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}\]

Hence: \[e^x \approx \sum_{n=1}^\infty \frac{x^n}{n!}\]