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

6.6 Sequences arising from the iterative solution of non-linear equations 219
Each of these series converges rapidly when x is small, and so can be used to obtain
usefulapproximations. Inparticular, wenote that
Ifxis smalland measured inradians
sinx≈x and cosx≈1− x2
2!
These formulae areknown as thesmall-angle approximations.
EXERCISES6.5
1 Thepower seriesexpansion ofe x is given by
e x =1+x+ x2
2! + x3
3! +···
andis valid foranyx.Take four terms ofthe series
whenx=0, 0.1,0.5and1, tocompare the sumtofour
termswith the value ofe x obtained fromyour
calculator. Comment upon the result.
2 Using the power series expansion forcosx:
(a) Writedown the power series expansion for
cos2x.
(b) Writedown the power series expansion for
cos(x/2).
Byconsidering the power series expansionfor
cos(−x) showthatcosx = cos(−x).
3 Byconsidering the power series expansionofe x find
∑ ∞k=0
1/k!.
4 Obtainacubicapproximation to e x sinx.
5 (a) Statethe power series expansion fore −x .
(b) Byusingyoursolution to (a)andthe expansion
fore x ,deduce the power seriesexpansions of
coshx andsinhx.
Solutions
1
x e x Sum to 4 terms
0 1 1
0.1 1.1052 1.1052
0.5 1.6487 1.6458
1 2.7183 2.6667
Values are in close agreementwhenxissmall.
2 (a) 1−2x 2 + 2x4
3
3 e
−··· (b) 1−x2
8 + x4
384 −···
4 x+x 2 + x3
3
5 (a) 1−x+ x2
2! − x3
3! +···
(b) coshx=1+ x2
2! + x4
4! +···,
sinhx=x+ x3
3! + x5
5! +···
6.6 SEQUENCESARISINGFROMTHEITERATIVESOLUTION
OFNON-LINEAREQUATIONS
Itisoften necessarytosolve equationsofthe form f (x) = 0.For example,
f(x)=x 3 −3x 2 +7=0, f(x)=lnx− 1 x = 0
To solve means to find values of x which satisfy the given equation. These values are
knownasroots.Forexample,therootsofx 2 −3x +2 = 0arex = 1andx = 2because