Find the Maclaurin series generated by...

Question

Find the Maclaurin series generated by the function f(x) = 3x^2 + 2x + 1

Answer 1

The Maclaurin series is a Taylor series expanded about x=0.

f(x) = f(0)(x - 0)⁰/0! + f'(0)(x - 0)¹/1! + f''(0)(x - 0)²/2! + ....

Terms with f''' and higher derivatives are zero, so

f(x) = f(0)x⁰/0! + f'(0)x¹/1! + f''(0)x²/2!

f(0) = 1

f'(0) = (6x + 2)|₀ = 2

f''(0) = 6

f(x) = 1(1)/1 + 2x/1 + 6x²/2 = 1 + 2x + 3x²

The Maclaurin series just returns the exact same polynomial.