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. Please submit your assignment as a R-Markdown document.
The formula for a Taylor Series centered about \(c\) is \[\begin{gather*} f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!} + \cdot\cdot\cdot \\ = \sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}x^n \end{gather*}\]
From Maclaurin Series, we obtain:
\[\begin{gather*} 1 + x + x^2 + x^3 + \cdot\cdot\cdot \\ = \sum_{n = 0}^{\infty}x^n \ \ \ x \in (-1, 1) \end{gather*}\]
Centered about 0:
\[\begin{gather*} 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdot\cdot\cdot \\ = \sum_{n = 0}^{\infty} \frac{x^n}{n!} \ \ \ x \in \mathbb{R} \end{gather*}\]
Centered about 0:
\[\begin{gather*} 0 + x - \frac{1}{2!}x^2 + \frac{2}{3!}x^3 - \frac{6}{4!}x^4 + \cdot\cdot\cdot \\ = \sum_{n = 0}^{\infty}(-1)^{n+1}\frac{x^n}{n} = \sum_{n = 0}^{\infty}(-1)^{n-1}\frac{x^n}{n} \ \ \ x \in (-1, 1] \end{gather*}\]