Basic Mathematics20805 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 $\frac{1}{x}$? [5]
- Function: $f(x) = \dfrac{1}{x}$ - Center: $a = 2$ $$f(x) = x^{-1}$$ $$f'(x) = -x^{-2}$$ $$f''(x) = 2x^{-3}$$ $$f'''(x) = -6x^{-4}$$ General pattern: $$f^{(n)}(x) = (-1)^n , n! , x^{-(n+1)}$$ $$f^{(n)}(2) = (-1)^n , n! , 2^{-(n+1)} = \frac{(-1)^n , n!}{2^{n+1}}$$ Values: -