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

218 Chapter 6 Sequences and series
(
)
10 (a) 1+4x 2 +6x 4 +4x 6 +x 8
(b) 1+ 4 x 2 + 6 x 4 + 4 x 6 + 1 (b) x −1/2 1 + x 2 − x2
8 + x3 valid for |x| < 1
16
x 8
12 (a) 1 −4x 2 +10x 4 −20x 6 validfor |x| < 1
11 (a) 1+ 1 2x − 1
8x 2 + 1
(
·)
16x 3 valid for |x| > 1 (b) x −8 1 − 4 x 2 + 10
x 4 − 20
x 6 · · valid for |x| > 1
6.5 POWERSERIES
Aparticularlyimportantclassofseriesareknownaspowerseriesandtheseareinfinite
series involving integer powers ofthe variablex. For example,
and
1+x+x 2 +x 3 +···
1+x+ x2
2 + x3
6 +···
arebothpowerseries.Notethatapowerseriescanberegardedasaninfinitepolynomial.
Many common functions can be expressed intermsof a power series,forexample
sinx=x− x3
3! + x5
− ··· x inradians
5!
which converges for any value ofx. For example,
sin(0.5) = 0.5 − (0.5)3
6
+ (0.5)5
120 −···
Taking justthe first threeterms,we find
sin(0.5) ≈ 0.5 −0.0208333 +0.0002604 = 0.4794271
as compared with the truevalue, sin0.5 = 0.4794255.
More generally, a power series is only meaningful if the series converges for the
particular value of x chosen. We define an important quantity known as the radius of
convergence,R, as the largest value for which anxchosen in the interval −R <x<R
causes the series toconverge.
Theopeninterval (−R,R)isknownastheintervalofconvergence.Testsforconvergenceofapowerseriesarethesubjectofmoreadvancedtexts.Furtherconsiderationwill
be given to power series in Chapter 18, but for future reference we give some common
expansions now:
sinx=x− x3
3! + x5
− ··· x inradians
5!
cosx=1− x2
2! + x4
− ··· x inradians
4!
e x =1+x+ x2
2! + x3
3! + x4
4! +···
all of which converge for any value ofx.