This week, we’ll work out some Taylor Series expansions of popular functions.
• \(f(x) = \frac{1}{(1-x)}\)
• \(f(x) = e^x\)
• \(f(x) = ln(1 + x)\)
• \(f(x) = x^{1/2}\)
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.
• \(f(x) = \frac{1}{(1-x)}f(0) = 1\)
• \(f'(x) = \frac{1}{(x-1)^2}f'(0) = 1\)
• \(f''(x) = -\frac{2}{(x-1)^3}f''(0) = 2\)
• \(f'''(x) = \frac{6}{(x-1)^4}f'''(0) = 6\)
Pattern is \(1 + x^2 + x^3 + x^4\)
\[\sum_{n=0}^{\infty}x^n\]
• \(f(x) = e^x f(0) = 1\)
• \(f'(x) = e^x f'(0) = 1\)
• \(f''(x) = e^x f''(0) = 1\)
• \(f'''(x) = e^x f'''(0) = 1\)
\[\sum_{n=0}^{\infty}\frac{x^n}{n!}\]
• \(f(x) = ln(1 + x)f(0) = 0\)
• \(f'(x) = \frac{1}{1 + x}f'(0) = 1\)
• \(f''(x) = -\frac{1}{(1 + x)^2}f''(0) = -1\)
• \(f'''(x) = \frac{2}{(1 + x)^3}f'''(0) = 2\)
• \(f''''(x) = -\frac{6}{(1 + x)^3}f''''(0) = -6\)
\[\sum_{n=0}^{\infty}\frac{-1^{n+1}x^n}{n}\]
• \(f(x) = x^{1/2}f(0) = 0\)
• \(f'(x) = x^{1/2}f'(0) = undefined\)