Taylor and Maclaurin polynomials

Problem 7.138 · hard

1. p_n(x) = Σ f⁽ᵏ⁾(0)/k! · xᵏ, for k = 0 to n.
   Reviewed
2. \[ \left. e^{x} \right|_{\substack{ x=0 }} = 1 \]
   f⁽0⁾(0).✓ Proved
3. \[ \left. \frac{d}{d x} e^{x} \right|_{\substack{ x=0 }} = 1 \]
   f⁽1⁾(0).✓ Proved
4. \[ \left. \frac{d^{2}}{d x^{2}} e^{x} \right|_{\substack{ x=0 }} = 1 \]
   f⁽2⁾(0).✓ Proved
5. \[ \left. \frac{d^{3}}{d x^{3}} e^{x} \right|_{\substack{ x=0 }} = 1 \]
   f⁽3⁾(0).✓ Proved
6. \[ x^{3} \cdot 1 \cdot \frac{1}{6} + x^{2} \cdot 1 \cdot \frac{1}{2} + x 1 + 1 = \frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1 \]
   Assemble the polynomial.✓ Proved