Find a formula for the \(n^{th}\) term of the Taylor Series of f(x), centered at c, by finding the coefficients of the first few powers of x and looking for a pattern.
\(f(x) = e^{-x};\) … \(c = 0\)
\(f'(x) = -e^{-x}\) … \(f'(0) = -1\)
\(f''(x) = e^{-x}\) … \(f''(0) = 1\)
\(f'''(x) = -e^{-x}\) … \(f'''(0) = -1\)
\(f^{(4)}(x) = e^{-x}\) … \(f^{(4)}(0) = 1\)
\(f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...\)
\(= 1 + \frac{-1}{1!}x + \frac{1}{2!}x^2 + \frac{-1}{3!}x^3 + \frac{1}{4!}x^4 + ...\)
\(\Sigma^\infty_{n=0} (-1)^{n} \frac{x^n}{n!}\)