Section 11.8 overview. We start by examining a couple of approximations for familiar functions:

Plot[{1/(1-x),
1,
1+x,
1+x+x^2,
1+x+x^2+x^3,
1+x+x^2+x^3+x^4,
1+x+x^2+x^3+x^4+x^5,
1+x+x^2+x^3+x^4+x^5+x^6
},{x,-1,1}]

Plot[{Exp[x],
1,
1+x,
1+x+x^2/Factorial[2],
1+x+x^2/Factorial[2]+x^3/Factorial[3],
1+x+x^2/Factorial[2]+x^3/Factorial[3]+x^4/Factorial[4]},{x,-2,2}]

Series[Exp[x],{x,0,5}]
f[x_]=Normal[%]
N[f[1]]
N[Exp[1]]
Plot[{Exp[x],f[x]},{x,-2,2}]

Series[Sin[x],{x,0,5}]
f[x_]=Normal[%]
f[1]
Plot[{Sin[x],f[x]},{x,-Pi,Pi}]

Series[Cos[x],{x,0,5}]
f[x_]=Normal[%]
f[1]
Plot[{Cos[x],f[x]},{x,-Pi,Pi}]