This week, we'll work out some Taylor Series expansions of popular functions.
\(f(x) = \frac{1}{(1-x)}\)
\(f(x) = e^x\)
\(f(x) = \ln(1 + x)\)
For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion.
When x = 0
\(1 + x + x^2 + x^3 + x^4 + x^5 + \cdot\cdot\cdot \\\)
\(\frac{1}{(1-x)} = \sum_{n = 0}^{\infty}x^n \ \ \ x \in (-1, 1)\)
When x = 0
\(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdot\cdot\cdot \\\)
\(e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!} \ \ \ x \in \mathbb{R}\)
When x = 0
\(0 + x - \frac{1}{2!}x^2 + \frac{2}{3!}x^3 - \frac{6}{4!}x^4 + \cdot\cdot\cdot \\\)
\(\ln(1 + x) = \sum_{n = 0}^{\infty}(-1)^{n-1}\frac{x^n}{n} \ \ \ x \in (-1, 1]\)