3.3
TAYLOR PoLY SEAIES
1-10-11-
Cnfa) > ydliddted by ofe rentidting Fa)
n see d pdttern.
evon yalidfor n'0 (o!1)
Theore m
Tf f hay d power series represen tati0n dF a that is,
fal 2 n (a-q) -4 14R
19 9
Then its toeft are 9 ven by the formyld.
Ch'fh)(u)
pito 999b
Thecre mn 9709
Sub Cn batk n+0, Senes We gef
Taylors i
ft) FR (a)( a - )
spettal case 9 : 9 i9
-0 Taylor series become x 9 o
f() p(h Maclaurin series.
Exqmple
Find Maclaurin series f f « ) *e * ard is radiuf of conv
f(a) TO
f)e f() e - fer all n
oO.0
, calrulated. = I
o
Rddius of conve rgo ne b9fbiEy
ud
c
(et h ' n h 9HDq 992
nT
Ratio Test
.
ns
On (nr)n g19halstwa
Jo by Rtio Tet ser(on vPigj sfer,.al iai
under which crum.stdhtes_is f(a): Z Taylor series.
9: ga f()
n
¢a-a)n9T
Tn () 2 )(2-4)
)0
i
n-th degree Taylor polynomidl of f@d
Theorem
1f fiz): Tn (a)+ Rn ()where Th the n-th deyree poly-
nonmio of f t q dro
im R)0
h
fer lskthen f s eudl to the sum afisTaylor sene
on Fhe ntervdl
Helpf-kow
14ylot' inequdlity
| ZE Ch ( 2 ) |4 M fer Ix- a & d ,then the remairder
n()f The Tuyorgeries setis fies_Hhe_inequaliy
N9 aUbA
R () M /2-/h/
fer z-al 4 d
(nt
TAYLOR PoLY SEAIES
1-10-11-
Cnfa) > ydliddted by ofe rentidting Fa)
n see d pdttern.
evon yalidfor n'0 (o!1)
Theore m
Tf f hay d power series represen tati0n dF a that is,
fal 2 n (a-q) -4 14R
19 9
Then its toeft are 9 ven by the formyld.
Ch'fh)(u)
pito 999b
Thecre mn 9709
Sub Cn batk n+0, Senes We gef
Taylors i
ft) FR (a)( a - )
spettal case 9 : 9 i9
-0 Taylor series become x 9 o
f() p(h Maclaurin series.
Exqmple
Find Maclaurin series f f « ) *e * ard is radiuf of conv
f(a) TO
f)e f() e - fer all n
oO.0
, calrulated. = I
o
Rddius of conve rgo ne b9fbiEy
ud
c
(et h ' n h 9HDq 992
nT
Ratio Test
.
ns
On (nr)n g19halstwa
Jo by Rtio Tet ser(on vPigj sfer,.al iai
under which crum.stdhtes_is f(a): Z Taylor series.
9: ga f()
n
¢a-a)n9T
Tn () 2 )(2-4)
)0
i
n-th degree Taylor polynomidl of f@d
Theorem
1f fiz): Tn (a)+ Rn ()where Th the n-th deyree poly-
nonmio of f t q dro
im R)0
h
fer lskthen f s eudl to the sum afisTaylor sene
on Fhe ntervdl
Helpf-kow
14ylot' inequdlity
| ZE Ch ( 2 ) |4 M fer Ix- a & d ,then the remairder
n()f The Tuyorgeries setis fies_Hhe_inequaliy
N9 aUbA
R () M /2-/h/
fer z-al 4 d
(nt
