TAYLOR SERIES APPOXIMATIONS - ASSIGNMENT 14

Work out the Taylor series approximations by hand for some common functions.

1. \(f(x) = \frac{1}{(1-x)}\)

\[ f(x) = \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\]

\[ f(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...\]

2. \(f(x) = e^x\)

\[ f(x) = e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

\[ f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + ...\]

3. \(f(x) = ln(1 + x)\)

\[ f(x) = ln(1 + x) = \sum_{n=0}^{\infty} (-1)^{n+1}\frac{x^n}{n}\]

\[ f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + ...\]