In Exercises 7 – 12, find a formula for the nth 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 patt�ern. (The formulas for several of these are found in Key Idea 8.8.1; show work verifying these formula.)
\[f(x)=e^{−x}; c=0\] \[f(0) = 1\] \[f'(x)= -e^{−x}\] \[f'(0) = -1 \]
\[f''(x)= e^{−x}\] \[f''(0) = 1 \]
\[f'''(x)= -e^{−x}\] \[f'''(0) = -1 \]
\[f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}\] \[f(x) = f(0) + \frac{f'(0)x}{1!} + \frac{f''(0)x^{2}}{2!} + ... + \frac{f^{n}(0)x^{n}}{n!}\]
\[f(x) = 1 - x + \frac{x^{2}}{2} + ...\] \[ = \sum_{n=0}^{\infty}\frac{x^{(n)}(-1)^{n}}{n!}\]