Taylor Polynomials and Series

The basic command to generate a Taylor series, in this case for sin(x) with terms through x^11. The extra “O[x]^12” indicates that the difference between these terms and sin(x) is less than some constant times x^12.

Series[Sin[x], {x, 0, 11}]

If you wish to have only the polynomial, without the extra “O[x]^12”:

Series[Sin[x], {x, 0, 11}]//Normal

This two part command plots both sin(x) and the Taylor polynomial for sin(x) through x^11 on the same axes, with x from -10 to 10 and the vertical scale from -2 to +2:

polyn[x_]=Series[Sin[x], {x, 0, 11}]//Normal;Plot[{polyn[x],Sin[x]},{x,-10,10},PlotRange->{-2,2}]