Basic Mathematics207710 marksNumerical: Taylor seriesTaylor series
Find the Taylor's series generated by f(x) = frac1x at a = 2. Where, if anywhere, does the series converge to frac1x?
Find the Taylor's series generated by $f(x) = \frac{1}{x}$ at $a = 2$. Where, if anywhere, does the series converge to $\frac{1}{x}$? [10]
- Function: $f(x) = \dfrac{1}{x}$ - Center of expansion: $a = 2$ Derivatives of $f(x) = x^{-1}$ $$f'(x) = -x^{-2}, \quad f''(x) = 2x^{-3}, \quad f'''(x) = -6x^{-4}$$ General form: $$f^{(n)}(x) = (-1)^n , n! , x^{-(n+1)}$$ Evaluate at $a = 2$ $$f(2) = \frac{1}{2}, \quad f'(2) = -\frac{1}{4}, \qu...