Data 605 Discussion 14

Exercises 3 - 6, chapter 8.8

\(f(x) = e^x; \, c = 0\) \ The Taylor series expansion for \(e^x\) around \(c = 0\) is given by: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \] Let’s find the first few terms of this series and identify the pattern: \[ f(x) = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots \] The pattern here is evident: each term is \(\frac{x^n}{n!}\).

\(f(x) = \sin(x); \, c = 0\) \ The Taylor series expansion for \(\sin(x)\) around \(c = 0\) is given by: \[ \sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \] Let’s find the first few terms of this series and identify the pattern: \[ f(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \] The pattern here is that each term alternates in sign and involves odd powers of \(x\), with the coefficient being \(\frac{1}{(2n+1)!}\).

\(f(x) = \frac{1}{1 - x}; \, c = 0\) \ The Taylor series expansion for \(\frac{1}{1 - x}\) around \(c = 0\) is given by: \[ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \] Let’s find the first few terms of this series and identify the pattern: \[ f(x) = \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \dots \] The pattern here is that each term is simply \(x^n\).

\(f(x) = \tan^{-1}(x); \, c = 0\) \ The Taylor series expansion for \(\tan^{-1}(x)\) around \(c = 0\) is given by: \[ \tan^{-1}(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \] Let’s find the first few terms of this series and identify the pattern: \[ f(x) = \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots \] The pattern here is that each term alternates in sign and involves odd powers of \(x\), with the coefficient being \(\frac{1}{2n+1}\).