1. Taylor’s Theorem (See Introduction to Real Analysis by Bartle and Sherbert for proves)

2. Applications For practice we denote
Find the Taylor expansion of sin(x) about
answer:

3. How about the Taylor expansion of Tan(x) at ?
The Taylor expansion of a function about
is called the Maclaurin series. Thus, Maclaurin series for sin(x) is,
It is impossible in practical applications to include an infinite number of terms. Therefore, the Taylor series has to be truncated after a certain order term

4. If the Taylor series is truncated after the N th order term, it is expressed as
Since cannot be found exactly, the error term is often approximated by

5. Example 1:
Find the Taylor expansion of about
which use the first three, four and five terms and evaluate for x= 0.5, respectively.
Answer:

7. Example 2:
Find the Taylor expansion of about
which use the first three, four and five terms and evaluate for x= 0.5, respectively.
Answer:

9. Summary from example 1 and 2:
Order of Trunc. Er.
Ratio
3.4
6.66
e-04
13.13
e-0.5
26.03