Answer:
f(x) = 1 + x + (x²/2!) + (x³/3!) + ....... = Σ (xⁿ/n!) (Summation from n = 0 to n = ∞)
Step-by-step explanation:
f(x) = eˣ
Expand using first Taylor Polynomial based around b = 0
The Taylor's expansion based around any point b, is given by the infinite series
f(x) = f(b) + xf'(b) + (x²/2!)f"(b) + (x³/3!)f'''(b) + ....= Σ (xⁿfⁿ(b)/n!) (Summation from n = 0 to n = ∞)
Note: f'(x) = (df/dx)
So, expanding f(x) = eˣ based at b=0
f'(x) = eˣ
f"(x) = eˣ
fⁿ(x) = eˣ
And e⁰ = 1
f(x) = 1 + x + (x²/2!) + (x³/3!) + ....... = Σ (xⁿ/n!) (Summation from n = 0 to n = ∞)
