Discussion 14

This document presents the Taylor Series expansions for the given functions.

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

This function is valid for \(|x| < 1\). The Taylor Series expansion for \(f(x)\) is:

\[ f(x) = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots \]

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

This function is valid for all real numbers \(x\). The Taylor Series expansion for \(e^x\) is:

\[ f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \]

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

This function is valid for \(-1 < x \leq 1\). The Taylor Series expansion for \(\ln(1 + x)\) is:

\[ f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \]

4. \(f(x) = \sqrt{x}\)

This function is valid for \(0 \leq x\). The Taylor Series expansion for \(\sqrt{x}\) is:

\[ f(x) = \sum_{n=0}^{\infty} \binom{1/2}{n} (-1)^n (x-1)^n \]

where \(\binom{1/2}{n}\) is the binomial coefficient.

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