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WTW238: Theme 3 (LU 3.3 -3.4) Summary $8.74   Add to cart
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WTW238: Theme 3 (LU 3.3 -3.4) Summary
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  • Course
  • WTW 238 - Mathematics (WTW238)
  • Institution
  • University Of Pretoria (UP)

This document contains a summaru of LU 3.3 - 3.4: Radius of convergence, Ordinary and Singular points, Maclaurin series, Taylor series, Taylor's inequality, Binomial series.

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Document information

  • December 24, 2022
  • 8
  • 2022/2023
  • Summary

Subjects

  • power series
  • 238
  • wtw238
  • taylor polynomials
  • polynomials
  • taylor
  • maclaurin series

Written for

  • University of Pretoria (UP)
  • WTW 238 - Mathematics (WTW238)
All documents for this subject (6)

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Content preview

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

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