Taylor Series Builder: Coefficients from Derivatives

BCMaclaurin series (Taylor series centered at a = 0)

f(x) = e^xP_{3}(x)

Degreen = 3

Maclaurin polynomial, degree 3

P_{3}(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}

Converges to f(x) = e^x on (-\infty, \infty).

f(0.5)

1.6487

P_3(0.5)

1.6458

error

0.0029

Every coefficient is c_k = \dfrac{f^{(k)}(0)}{k!}. That is why raising the degree adds one more term and pulls P_{3} closer to the curve.

k	f^{(k)}(0)	k!	c_k = f^{(k)}(0)/k!
0	1	1	1
1	1	1	1
2	1	2	\frac{1}{2}
3	1	6	\frac{1}{6}
4	1	24	\frac{1}{24}