[Calculus] Taylor series

Question

The Taylor series for f(x)=e^x at a = 3 is ∑∞_n=0 c_n(x-3)ⁿ

Find the first few coefficients.

C₀=

C₁=

C₂=

C₃=

C₄=

Answer 1

f(3) = e³; f'(x) = f''(x) = f''(x) = f⁽⁴⁾(x) = e^x. f'(3) = f''(3) = f'''(3) = f⁽⁴⁾(3) = e³.

C₀ = e³;

C₁ = e³/1! = e³;

C₂ = e³/2! = e³/2

C₃ = e³/3! = e³/6;

C₄ = e³/4! = e³/24.

Answer 2

f(x) = e^x

f(x) = f'(x) = f"(x) = f'''(x), ...

C₀ = f(3) = e³

C₁ = f'(3) = e³

C₂ = f"(3) / 2! = e³ / 2

C₃ = f'''(3) / 3! = e³ / 6