en bentle
Power series.
f ft 4 nea ”
mm ot ae (xe)
(xc) is general form of @ power series.
cw called the center of the power series
THEDREM torany power series S29 iy (4 —" one of the following aematives must hokt
Conve
69) the series may co
Bh Gin te series nay conver
Ci) tere may exist a postive eal nbersich that the avis converges
very tatsfying rel Rand dvergesat every x stetyingie el > Me
Tnihiscane he sein may a yn conver aco he to endpoint
Rann ct
tn enh ofthese cases ie Sonvergece is abilue excep possibly atthe endpoints
‘Rand x — 6+ R incase (i), ia) Convergent for |x-cl
R xEtR
May er may not be convergent when x=c-R and xceR
js called tte radus of convergence,
J
(e-R, ORgeLe-R, eR prCe~ RyetRT gg Le-R, ca RI to an interval of
Conyerg ence
by Rots Test, Gruen 24 (x-c)”
wo
ae
- (ee)
lien [Peete = Vien [Pot] [x-e] <4
tm [ay (x-c)*| Cae Fon
L
Ix-cl <4. 2 Q SD The sees u
L convergent:
Determine the centre, ras, and interval of convergence ofeach 4
of the power series in xercnes I-A S a x-0)
2 Sone Le y
2. StS
ara
1 Zoe Ss @) ows
eae as center
on oP a
lean ea) as 4on ix 4] = OS Coneagerh
no \ a cx 2y | sa Cat i Feralas
ae 5 es
Vv
Ma tpt? hs, Caleb ree
vou s
°
ake xr) UM gy 2S Got
Xoo Be he
@ x a
ec
Radive of convergence is 00
Taterval of convergence & (-0,00) alin Gays Reco Test: (For poilne eenes) en
1) OL LEA > Convergent
2) LOA D> Bivergeat
3) L=4 2 No informetion
lim 4 eb
Rae An
(x2) D> = 2 ie the center
Gx-t-»)"
lxt2] = 4 heal
lx
z
ae
A lxqel ad D [x2] <2 D Convergent ¢ eocl 2S xen KL B HKK <0
e 1 4
x=-4 Sah (-2) = (-1)
mA
net
Ys eanvegent once Qe aw
decceasing and lim = 0 (By Attermatry Series Test) By Rebs Test the series is convergent when elx-4]) <4
Ss al <(S) SDS Ret nw He radus of convergence
@
R
> thd x-4 y-hox cath
€ e €
xe4-t > Ss é ey (sy 3 fOr AL is convergent (p23>4)
by p—dest.
is converged by Allerrach ng
Sens Test sme A ag, is decreasing lim 4-0.
oe
Interval of Convergence ts Cet. +4]
4
Ex ase +2 2)
fx ® 2) +(
a
ed nite Tat 4
Li Compe S : + a
lies OD Bom ESbn aed Ss
N00
ere. beth divecgort
or beth convergent meee, 2 ob Mem eg sa 2 Se aeenp
ann 2 os L Co. net 3
tot) Biv lig aq =k
130
OSL ct d Com L>1 dw 0 L
ion =
1200 an
Of L <1 5 Com
L>4 5 bv.
at der Yl 4-0 ay Conv.
no Mow nt
eH.
Serves
Test
linn [Onl +O SD lim Aq FO
“: A300
ken CD £0 > biver gent.
neco eA By Bier gene
Tet
oo
L )
5
a= 8
TT HF ota ngage cs 7x with center O
n=O
converges when -4< x <4 and
a4
AL KLID = oe x
n=0
Differentiation tnd Tntegrartion of Power Series
predate Ge ae
THEOREM Term-by-term aifferetaton and integration of per series
19
orga pagencncennne ae forwmnnd CRA se
10
utente batten, (Rar oF
by starting with the geometsie series
1 aSrateetteee (Chere d
=
and using differentiation, integration, and substitution, Where is each series valid?
«) fa)=e t= 3 K Noe exc
(a 2 nso
\ ey
i Se 2 ee en aA
F'ta) = (u-x) je 4. (4-x) aye
a. hr = <) Intex)
: x
(a dt = -An-+) | = = da t= 9) da d= ~ tala»)
A4-+ 9 :
0
2 i net
ic Z— D Inlt-x
~ lina! 25 rag : 5x
Replace x with -x
0 a ) “ Cal '2 i
| 2 on : - Wt ney
nA tx 22 AA
=> ned
«0
In (4x) = 2ark = S08". ia
ane
nz0 n+4
nz 0 October 8
K
wn eeennes
£)- GO Oren” {aje- oF
ef AT a HQ Fe
21 "alll
*
= Bw = tle aloe” +x? + -
Z u
me Qe a, a, \
(ero) " \ = fee)
fre) att) Ble) Se
21 $0) =4
Hols +. d>t'e=4
(x)
(x
$'(xd= ot (ex) e) =
ays 2 ae
w)
Ff &) 21
rs
DEFINITION | Taylor and Mactaurin series
8 | If f(x) has derivatives of all orders at x = € (ie. if (4%) exists for k =
041,2,3,+-dy then the seis
SIMO y ge
pee
= f+ F094 LO «7 +O
-eP + OO a
is catled the Taylor series of f about c (or the Taylor series of f in powers of
x6). Ife =0, the term Maclaurin series is usually used in place of Taylor Ex Find the Maclauin sedes of f(x) =sinx
$60) + Ftorx'+ £9 x4 lo) x? 4 -
2 3
Food = 9 ede cosx > Flo)= 1
$°(x) =~ sin x D $l) =O
F" (xX) = osx > f"l0) =~!
$M) = siox s $5) = 0
FW 2 cox FF) = 1
oo
' > ane
K-LK 4 xt ag = Ay x 7
a ts FH 7
Ko Gane a
Remo k
Not all function is equal +2 sts Tayler Series.
DEFINITION | Anatytic functions
‘A function f is analytieateif has 2 Taylor series a c and that series converges
to /(&) in an open interval containing e. If f is analytic at exch point of an open
serv hen we sty it is analytic on that interval
st aq—t
A) x
2n-1)] yo Find Taylor Series of e*
‘J
about c.
"
7) fet > Fee Dd HODES
ia) n
3 mc) =) Cee EC) cee ee NE:
ei
& Cx-e)" iy the Taylor series of ©) shoot ce.
al
ea
-ox ce
4. Ix-cl= 0 sb ves)
300 eee
For all xeIR
Let os Lo. © aoe 2
g(x) = 2 (xe) = et ke) + & xc) + &-(x-€) + --
ot 21 3!
gue e + ef Lt) + © Bx) 4 --
a AT2I ghed = et 8 Le) + LM re 4
paral 2q2I
7 x
wlan s ae) t+ 29H) 3 GH = C-€
a c 4
" (c)=e=Ce
) 3
ie >C="
co Se :
: DgWe=
e = (ee) & Ean gwee
nea enalyhcel functor.
o a
e = xX" 5 moclurin Sees of 2%
al
nee
oo 00
a anet
Sink = C1) x cos =
(ae) (anI
neo n=0 “EXAMPLE 3 Ob!in Maclaurin series forthe following Functions
(a) eX thy sn?) (©) sin? x.
.
oO oD
Df Seam. ;
e
al ——
oa n=0
al gor
. ant4 00 " 44
b) sinx = oo X > sin(x?) = (=. G*)
244)! :
eG aco CAnt4)
n 2 0
SS sink) _ ear Sy Xi
x Evaluate (a) lim usa
7 a0 x
Solution
. x — sine 0
ogee [3]