Introduction

Taylor Series Expansions

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

  • First 5 in the Series Expansion: \[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 \]

  • Taylor Series Expansion: \[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \]

  • Valid Range: \(|x| < 1\)

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

  • First 5 in the Series Expansion: \[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \]

  • Taylor Series Expansion: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

  • Valid Range: All \(x\)

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

  • First 5 in the Series Expansion: \[ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \]

  • Taylor Series Expansion: \[ \ln(1 + x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \]

  • Valid Range: \(-1 < x \leq 1\)

4. Function \(f(x) = x^{1/2}\)

  • First 5 in the Series Expansion: \[ x^{1/2} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 \]

  • Taylor Series Expansion: \[ x^{1/2} = \sum_{n=0}^{\infty} \binom{1/2}{n} x^n \]

  • Valid Range: \(x \geq 0\)