BOUNDING THE ERROR TERM IN TAYLOR’S THEOREM
Suppose we want to find a Taylor Polynomial that approximates e2x with an error
less than 10−8 for all values of x in the interval [−0.5, 0.5]. How many terms of the
Taylor series for e2x should we include?
Notice that x = 0 is at the center of the interval we are given. According to Taylor’s
Theorem there is a value of ξ between 0 and x such that
2n n 2n+1 e2ξ n+1
22 2 23 3
x + x + ··· +
x +
x
2!
3!
n!
(n + 1)!
where the last term is the error term. The smallest value of n for which the error
term is less than 10−8 is the degree of the lowest degree Taylor polynomial that
provides a suitable approximation. Thus, we seek the smallest n for which
(2x)n+1 e2ξ −8
(1)
(n + 1)! < 10
e2x = 1 + 2x +
for all x in [−0.5, 0.5]. For these values of x we observe that:
(1) |2x| ≤ 1 so |2x|n+1 ≤ 1,
(2) |ξ| < |x| so |2ξ| < 1 and, since e < 3, we know e2ξ < 31 = 3 (here we pick
the smallest whole number greater than e; other choices are possible).
Thus
(2x)n+1 e2ξ 3
(n + 1)! < (n + 1)!
and selecting n such that
3
< 10−8
(n + 1)!
guarantees that (1) is satisfied.
The table of values for n = 8 through n = 12 shows that n = 11 is sufficient:
n
3/(n + 1)!
8 8.26719577e-06
9 8.26719577e-07
10 7.51563252e-08
11 6.26302710e-09
12 4.81771315e-10
This means that the Taylor polynomial consisting of the first 12 terms (n = 0
through n = 11) of the Taylor series will provide sufficient accuracy.
Date: Written January 26, 2011; Revised January 2, 2015 and January 23, 2017.