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

18.6 Taylor and Maclaurin series 525
Aspecial,commonlyused,caseofaTaylorseriesoccurswhena = 0.Thisisknown
as theMaclaurin series.
Maclaurin series:
p(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2! +y(3) (0) x3
3! +···+y(n) (0) xn
n! +···
Example18.10 Determinethe Maclaurinseries fory = e x .
Solution Inthisexampley(x) = e x and clearlyy ′ (x) = e x too.Similarly,
y ′′ (x) =y (3) (x) = ··· =y (n) (x) =e x
forall values ofn. Evaluating atx = 0 yields
and so
y(0)=y ′ (0)=y ′′ (0)=···=y (n) (0)=1
p(x)=1+x+ x2
2! + x3
3! +···+xn n! +···
Asmentionedearlier,theseriesandthegeneratingfunctionareequalforallvaluesofx.
Hence,
e x =1+x+ x2
2! + x3
3! +···
forall values ofx, that is
∞∑
e x x n
=
n!
n=0
Example18.11 Obtain the Maclaurinseries fory(x) = sinx.
Solution y(x)=sinx, y(0)=0
y ′ (x)=cosx, y ′ (0)=1
y ′′ (x)=−sinx, y ′′ (0)=0
y (3) (x)=−cosx,
y (3) (0)=−1
y (4) (x)=sinx, y (4) (0)=0
p(x) =y(0) +y ′ (0)x + y′′ (0)x 2
2!
=x − x3
3! + x5
5! − x7
7! +···
∞∑ (−1) i x 2i+1
=
(2i +1)!
i=0
+ y(3) (0)x 3
3!
+···