082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

524 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
3 3.435 ×10 −2
4 (a) − t3
16 + t2 2 − 3t
2 +2
(t −2) 4
(b)
c 5 whereclies between 2 andt
(c) 0.5
5 (a) 5c(1 +3c)(x −1) 4 forcbetween1andx
(b) 70
18.6 TAYLORANDMACLAURINSERIES
We have seen that y and its first n derivatives evaluated at x = a match p n
(x) and its
firstnderivatives evaluated atx = a. We have studied the difference betweeny(x) and
p n
(x), thatisthe errorterm,R n
(x).
AsmoreandmoretermsareincludedintheTaylorpolynomial,weobtainaninfinite
series,called aTaylorseries. We denotethisinfiniteseries by p(x).
Taylor series:
p(x) =y(a) +y ′ (a)(x −a) +y ′′ (x −a)2
(a)
2!
+···+y (n) (x −a)n
(a) +···
n!
+y (3) (x −a)3
(a)
3!
ForsomeTaylorseries,thevalueoftheseriesequalsthevalueofthegeneratingfunction
for every value ofx. For example, the Taylor series for e x , sinx and cosx equal the
valuesofe x ,sinxandcosxforeveryvalueofx.However,somefunctionshaveaTaylor
series which equals the function only for a limited range of x values. Example 18.14
gives a case of a function which equals its Taylor series only when −1 <x<1.
To determine whether a Taylor series, p(x), is equal to its generating function,y(x),
we need to examine the error term of ordern, that isR n
(x). We examine this error term
as more and more terms are included in the Taylor polynomial, that is as n tends to
infinity. If this error term approaches 0 asnincreases, then the Taylor series equates to
thegeneratingfunction,y(x).Sometimestheerrortermapproaches 0asnincreasesfor
all values ofx, sometimes it approaches 0 only whenxlies in some specified interval,
say, for example, (−1,1). Hence wehave:
IfR n
(x) → 0 asn → ∞ for all values ofx, then the Taylor series, p(x), and the
generatingfunction,y(x), are equal forallvalues ofx.
IfR n
(x) → 0asn → ∞forvaluesofxintheinterval (α,β),thentheTaylor
series,p(x),andthegeneratingfunction,y(x),areequalforxvaluesontheinterval
(α,β). For values ofxoutside the interval (α,β), the values of p(x) andy(x) are
different.
By examining the error terms associated withy = e x ,y = sinx andy = cosx it is
possible to show that these errors all approach 0 asn → ∞ for all values ofx. Hence
the functionsy = e x ,y = sinx andy = cosx are allequal totheircorrespondingTaylor
series forall values ofx.