Problem :
Approximate e^{0.1} to within 10^{3} using Taylor's formula.
The error term in Taylor's formula is
r_{n}(x) = x^{n} = x^{n} < x^{n} 

where
c is some number in the interval
(0, 0.1). Thus
r_{3}(0.1) < 1/3000 < 10^{3},
so it is sufficient to go up to the degree
2 term in approximating
e^{0.1}. We have
e^{0.1} 1 + 0.1 + = 1.105 

Problem :
Approximate sin( 0.1) to within 10^{5}. The Taylor series for sin(x) begins
The absolute value of the error term in Taylor's formula is
 r_{n}(x) = x^{n} =  x^{n}≤ 

Noting that
 r_{4}( 0.1)≤ 0.1^{4}/4!≤10^{5}, we see that we need to use up to the
degree
4 term in the Taylor series. We have
sin( 0.1) ( 0.1)  –0.09983 
