Write The Taylor Series

Question

Use sigma notation to write the Taylor series about x = x0 for the function. e^(−3x), x0 = −1/3.

Answer 1

f(x) = e^-3x

= f(-1/3) + f'(-1/3)(x - (-1/3)) + f"(-1/3) / 2! (x - (-1/3))² + f'''(-1/3) / 3! (x - (-1/3))³ + ...

= e - 3e(x + 1/3) + 9e / (2!) (x + 1/3)² - 27e / 3! (x + 1/3)³ + ...

= e∑_{(n = 0 to ∞)} [(-3)ⁿ(x + 1/3)ⁿ / n!]