Maclaurin series
Rebuild a function from its derivatives at zero, one power of x at a time.
f(x) = f(0) + f′(0)x + f″(0)x22! + f‴(0)x33! + …
The Maclaurin series builds a polynomial that copies the function at x = 0.
- f(0) finds the y-intercept, so it starts at the same point as the curve.
- f′(0)x finds the gradient. Now it is the tangent at x = 0.
- The x2 term matches the second derivative, so the series curves with the function.
- Each term after that matches the next derivative, so it stays on the curve for longer.
Terms
sin x≈x
The general formula is
(−1)kx2k+1(2k+1)!
x
1
real value
0.8415
approximation
1
percentage error
18.84%