f[x_]:=Sqrt[x]
a=4

t3[x_]:=f[a]+f'[a](x-a)+f''[a]/2!(x-a)^2+f'''[a]/3!(x-a)^3

t3[x]

t10[x_]=Normal[Series[f[x],{x,a,10}]]

d=50
Plot[{f[x],t10[x]},{x,a-d,a+d}]

Plot[{f[x]-t10[x]},{x,a-d,a+d}]

{f''''[4],t3''''[4]}

r3[x_]:=f[x]-t3[x]

Plot[Abs[r3[x]],{x,2,6}]

m=.09
m/4!(2)^4

tolerance: .0001

Abs[f'''''''[2]]/(6+1)!(2.0)^(6+1)

n=60
Abs[Derivative[n+1][f][2]]/(n+1)!(2.0)^(n+1)

f[x_]:=Log[x]
n=6
Abs[Derivative[n+1][f][1]]/(n+1)!(0.3)^(n+1)