Mathematica code for Animating Taylor Series Expansion
Clear[f,p]
f[x_]=Sin[x] Sin[x]
(* poly[x,n] is an approximate polynomial of f[x] obtained by Taylor expansion of f[x],of order n, near x0 *)
Do[poly[x_,n]=Series[f[x],{x,0,n}]//Normal,{n,1,9}]
poly[x,9] 3 5 7 9 x x x x x - -- + --- - ---- + ------ 6 120 5040 362880
(* f[x] is in red amd poly[x,n] is blue*)
Do[Plot[{f[x],poly[x,n]},{x,0,3Pi/2},PlotRange->{0,1.5}, Epilog->{{Text["n = ",{0.5,1.4}]},{Text[n,{.65,1.4}]}}, PlotStyle->{RGBColor[1,0,0],RGBColor[0,0,1]}],{n,0,9}]
RED=f(x)=sin(x)
BLUE=TaylorPolynomial of order n=1,2,3,4,5,6,7,8,9
Note:As shown in plot and the list below, for largr value of n=9, poly[x,n] becomes equal to f[x]
when the remainder converges to zero.
remainder[x_,n_]=f[x]-poly[x,n] -poly[x, n] + Sin[x]
-7 -7 -13 {0.00999983333416667, -1.66665833335744 10 , -1.66665833335744 10 , 8.33331667560167 10 , -13 -18 -18 8.33331667560167 10 , -1.734723475976807 10 , -1.734723475976807 10 , 0., 0., 0.}