Theorem(RemainderEstimation)
If |f(n+1)(x)| ≤Mfor |x-x 0 | ≤r, then we havethe followingboundforRn(x):
for |x-x 0 | ≤r.Example 1 The functionexis equalto its Maclaurinseriesfor allx. Proof.Letf(x) =ex. We needto find
the aboveboundonRn(x).
If |x| ≤r,f(n)(x) =ex≤erforn≥ 0 and the remainderestimationgives for
|x| ≤r.
Since by the squeezeTheorem.
So. Henceexis equalto its Taylorseries for allx.
Example 2 (TruncationError)Whatis the truncationerrorof approximating by its third-
degreeMaclaurinpolynomialin for |x| ≤ 0.1.
Solution.
.For.
So. This is the truncationerrorof approximatingby
the third-degreeMaclaurinpolynomial.
Exercise
- Find the powerseriesrepresentationoff(x) =sin xatx= 0 for allx. Why is it the Maclaurinseries?
- Find the powerseriesrepresentationoff(x) =cos xat for allx. Whyis it the Taylorseriesat