The Taylor Series

The error term in Taylor's formula is

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

The absolute value of the error term in Taylor's formula is

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