Find the Taylor series of e^x about x = 0 up to the x^3 term

Problem: Find the Taylor series of $f(x) = e^x$ centered at $x = 0$ up to the $x^3$ term.

Steps

1. Compute the derivatives through order 3
   $$f(x) = e^x, \quad f'(x) = e^x, \quad f''(x) = e^x, \quad f'''(x) = e^x$$
2. Evaluate the function and its derivatives at $x = 0$
   $$f(0) = e^0 = 1, \quad f'(0) = 1, \quad f''(0) = 1, \quad f'''(0) = 1$$
3. Apply the Taylor series formula centered at $x = 0$
   $$P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3$$
4. Substitute the evaluated derivatives
   $$P_3(x) = 1 + (1)x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3$$
5. Simplify the factorial coefficients
   $$P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$