Taylor Series

Taylor and Maclaurin polynomial approximation of a function

Use x with + − * / ^ and smooth functions like sin, cos, tan, exp, ln, sqrt, sinh, atan.

Expand around this point. 0 gives the Maclaurin series.

Highest power of (x − a). A whole number from 0 to 12.

Compare the polynomial to the true value here. Leave blank to skip.

sin(x) ≈ x − 0.166667x³ + 0.00833333x⁵
Maclaurin series (a = 0)
Taylor polynomial Pₙ(x)
x − 0.166667x³ + 0.00833333x⁵
Pₙ(x) — approximation
0,9333333333
f(x) — actual
0,9092974268
Absolute error |f(x) − Pₙ(x)|
0,0240359065
Coefficients
kf⁽ᵏ⁾(a)cₖ = f⁽ᵏ⁾(a)/k!
000
111
200
3-1-0,1666666667
400
510,0083333333
The function and its Taylor polynomial about the center, which hug near a and separate as the polynomial diverges. f(x) = sin(x), n = 5.The function and its Taylor polynomial about the center, which hug near a and separate as the polynomial diverges. f(x) = sin(x), n = 5.xy-5-4-3-2-112345-1-0.50.51a
f(x)Pₙ(x)