IsT(x) convergentatx=a? Thereis no guaranteeexceptata=x 0. The secondquestionis:
IfT(x) convergesatx=a, doesit equalf(a)? The answeris negativeas showby the function:
Then
It can be verifiedthat.
So the Maclaurinseriesis 0, clearlydifierentfromfexceptatx= 0.
Nevertheless,here is a positiveresult.
TheoremIffhas a powerseriesrepresentationatx=x 0 , i.e.
for |x-x 0 | <Rc, then the coefficientsare givenby.So any powerseriesrepresentationatx=x 0 has the form:
Exercise
- Find the higherorderderivativesof the functionf 1 (x) abovethus recursivelyshowingf 1 (n)(0) = 0 forn≥
0 - Verify the Theoremusingterm-by-termdifierentiation.
TaylorsFormulawith Remainder, RemainderEstimation,TruncationError
Recallthe remainderR n(x) of thenth-degreeTaylorpolynomialatx=x 0 is givenbyRn(x) =f(x) -T
n(x).
Theorem(Convergenceof Taylorseries)
If for |x-x 0 | <Rc, thenfis equalto its TaylorSerieson the interval|x-x 0 | <Rc.
The abovecondition couldbe achievedthroughthe followingbound.