,:at's ,next8

  ,i want 65d ? c\rse ) j a qk glimpse
---b1 ,i hope1 a tantaliz+ "o---( :at
-es next1 ( ": "o mi<t g f "h4
  ,"o way 6?9k ab ! :ole project ( f9d+
tang5t l9es t has o3upi$ u ? seme/] is
6say t we've be5 try+ 6f9d ! /rai<t l9e
t be/ approximates y .k f(x) ne> ! po9t
(a, f(a))_4 ,if1 = 3v5i;e1 we take
a .k #0, !n we wd 2 look+ =a l9e ^:
value at #0 mat*$ ! value f(0) ( ;f at
#0, & ^: derivative at #0 mat*$ !
derivative f'(0) ( ;f at #0_4 ,! equ,n (
? l9e wd 2
  y .k f(0)+f'(0)x_4
  ,n[ let's ask ! next "q3 ,9/1d (a
/rai<t l9e1 :y n take a p>abola8 ,:at's
! p>abola t be/ approximates y .k f(x)
ne> x .k #0_8
  ,let \r p>abola look l
  y .k a+bx+cx^2_4
,we wd c]ta9ly want y(0) .k f(0), : m1ns
t a .k f(0)_4 ,we wd al want
                                      #j
y'(0) .k f'(0), : wd m1n t b .k f'(0)_4
,b n[ we cd al ask t y''(0) .k f''(0), :
wd m1n t #2c .k f''(0), or t
c .k ?f''(0)_/2#_4 ,! p>abola "!=e looks
l
  y .k f(0)+f'(0)x+?f''(0)_/2#x^2_4
,notice t ? is j \r be/ fit /rai<t l9e )
"o extra t]m a4$ on4
  ,x's n h>d 6?9k :at ! next /ep mi<t 24
,:at if we look =! cubic polynomial t
be/ approximates f(x) ne> x .k #0_8 ,if
we let
  y .k a+bx+cx^2"+dx^3,
!n ;a, ;b, & ;c wd 2 -put$ ! same way z
abv1 &! results wd 2 ! same4 ,n[1 h["e1
we cd al ask t y'''(0) .k f'''(0), : wd
say t #6d .k f'''(0), or t
d .k ?f'''(0)_/6#_4 ,! cubic curve is
"!=e
  y .k f(0)+f'(0)x+?f''(0)_/2#x^2"
    +?f'''(0)_/6#x^3_4
  ,if y keep play+ ? game1 us+ a f\r?
degree polynomial & try+ 6get ! "r #4th
derivative at #0, y get
                                      #a
  y .k f(0)+f'(0)x+?f''(0)_/2#x^2"
    +?f'''(0)_/6#x^3"
    +?f^(4)"(0)_/24#x^4,
an s on4
  ,a bit ( reflec;n %[s t all ! t]ms
we've -put$ h ! same =m4 ,! polynomial
we j -put$ c 2 writt5 z
  y .k f(0)+?f^(1)"(0)_/1&#x^1"
    +?f^(2)"(0)_/2&#x^2"
    +?f^(3)"(0)_/3&#x^3"
    +?f^(4)"(0)_/4&#x^4_4
,"o c ev5 write ! f/ t]m 9 ! same =m1 s
z 6get
  y .k ?f^(0)"(0)_/0&#x^0"
    +?f^(1)"(0)_/1&#x^1"
    +?f^(2)"(0)_/2&#x^2"
    +?f^(3)"(0)_/3&#x^3"
    +?f^(4)"(0)_/4&#x^4,
z l;g z we rememb] t x^0 .k #1 & t f^(0)
j m1ns f, & t ? fomula1 l _m o!rs1 "ws \
nicely if we def9e #0& 6be #1_4
  ,x doesn't take m* imag9,n 6see h[
63t9ue ? sum 6get m t]ms4

                                      #b
  ,! polynomials we're -put+ ? way >e
call$ ,taylor polynomials1 af ,brook
,taylor1 an ,5gli% ma!matician ! g5],n
af ,newton4 ,!y 7 al "kn 9 ,9dia
6,madahva 9 ! #14th c5tury4
  ,z a 3crete example1 x's n h>d 6"w \
^! polynomials explicitly =! func;n
f(x) .k sin x_4 ,= ? func;n we h
  f^(0)"(0) .k sin #0 .k #0
  f^(1)"(0) .k cos #0 .k #1
  f^(2)"(0) .k -sin #0 .k #0
  f^(3)"(0) .k -cos #0 .k -#1
  f^(4)"(0) .k sin #0 .k #0
& s on4 ,! sequ;e ( derivatives rep1ts )
p]iod #4_4 ,! f/ few ,taylor polynomials
= f(x) .k sin x "!=e 9clude
  y .k ?f^(0)"(0)_/0&#x^0"+?f^(1)"(0)_/
    1&#x^1
  .k x
  y .k ?f^(0)"(0)_/0&#x^0"+?f^(1)"(0)_/
    1&#x^1"+?f^(2)"(0)_/2&#x^2"+?f^(3)"(
    0)_/3&#x^3
  .k x-?x^3"_/3&#
  y .k ?f^(0)"(0)_/0&#x^0"+?f^(1)"(0)_/
                                      #c
    1&#x^1"+?f^(2)"(0)_/2&#x^2"+?f^(3)"(
    0)_/3&#x^3"+?f^(4)"(0)_/4&#x^4"+
    ?f^(5)"(0)_/5&#x^5
  .k x-?x^3"_/3&#+?x^5"_/5&#_4
,^! polynomials >e plott$ al;gside
f(x) .k sin x 9 ,figures #1, #2, #3,
resp4 ,notice t 9 a4i;n 6mat*+ ! s9e
func;n bett] & bett] "r at ! orig91 ^!
polynomials agree )! s9e func;n ov] bi7]
& bi7] 9t]vals c5t]$ at ! orig94
  ,! same ?+ "ws = cos x, except t cos x
3ta9s ev5 p[]$ t]ms 9/1d ( odd p[]$
t]ms4
  ,well1 ! next ?"\ is cle>1 "r8 ,:at if
9/1d ( a4+ up a f9ite numb] ( t]ms 6get
a ,taylor polynomial1 we j kept a4+ ="e
6get a ,taylor s]ies1
  ".,s%n .k #0<,=]?f^(n)"(0)_/n&#x^n_4
,x turns \ t 9 ! case (! s9e & cos9e
func;ns1 ? 9f9ite s]ies is actu,y equal
6! func;n "ey":4 ,9 o!r ^ws1 = e r1l
numb] ;x,
  sin x .k ?x_/1&#-?x^3"_/3&#+?x^5"_/5&#
    -?x^7"_/7&#+- '''
                                      #d
  cos x .k #1-?x^2"_/2&#+?x^4"_/4&#-
    ?x^6"_/6&#+- '''
,^! =mulas1 tgr ) "s trig id5tities1 >e
actu,y :at yr calculator uses 6-pute
trig func;ns4
  ,taylor s]ies give an ev5 simpl]
expres.n =! func;n e^x_4 ,we "k t if
f(x) .k e^x, !n f'(x) .k e^x .k f(x)_4
,=! same r1son1 = e non-negative 9teg] n
, f^(n)"(x) .k f(x)_4 ,s9ce
f(0) .k e^0 .k #1, x foll[s t = e
non-negative 9teg] ;n, f^(n)"(0) .k #1_4
,! ,taylor s]ies = e^x is ?us
  e^x .k #1+x+?x^2"_/2&#+?x^3"_/3&#
    +?x^4"_/4&#+?x^5"_/5&#+ '''
,ag1 x turns \ t ! equal;y is exact =
all r1l numb]s ;x_4
  ,^! s]ies repres5t,ns prove 6be v
p[];l tools = /udy+ func;ns4 ,want 6be
3v9c$ t ?d_/dx#sin x .k cos x_8 ,j take
! derivative (! ,taylor s]ies = sin x,
t]m 0t]m1 & y get ! ,taylor s]ies =
cos x_4 ,want 6-pute ! limit
"lim%x $o #0] ?sin x_/x#_8 ,use !
                                      #e
,taylor s]ies4
  "lim%x $o #0] ?sin x_/x#
    .k "lim%x $o #0] ,?x-?1_/3&#x^3"+?1
      _/5&#x^5"-?1_/7&#x^7"+ ''' ,_/x,#
    .k "lim%x $o #0] @(1-?1_/3&#x^2"+?1
      _/5&#x^4"-?1_/7&#x^6"+ '''@) .k #
      1_4
,s despite :at ,i sd e>li]1 y c factor
an ;x \ ( sin x_4
  ,i don't want 6imply t "ey?+ ,i've j
sd is trivial4 ,"! >e 9 fact a lot (
subtle "qs t >ise "h4 ,:5 does an 9f9ite
s]ies 3v]ge8 ,:5 is ! ,taylor s]ies
equal 6! orig9al func;n8 ,:5 is x
legitimate 6-pute ! derivative (a
,taylor s]ies 0di6]5tiat+ ea* t]m
sep>ately8 ,a ton ( 9t]e/+ "qs 9
ma!matics >e rais$ 0^! examples1 &
resolv+ !m l1ds 6_m deep & b1uti;l
4cov]ies4
  ,a f9al t1s] =! future4 ,a fam\s !orem
( ,eul] says t if i .k >-1], !n
  e^i.? .k cos .?+isin .?_4
,= 9/.e1 if .? .k .p, !n
                                      #f
  e^i.p .k cos .p+isin .p .k -#1,
or1 rewrit+ sli<tly1
  e^i.p"+1 .k #0_4
,x's be5 sd wryly t ? =mula is ! mo/
b1uti;l 9 all ma!matics1 9 "p 2c x 3ta9s
! #5 mo/ important numb]s 9 ma!matics1
once ea*4
  ,:at possi# s5se c "o make ( ,eul]'s
,!orem1 ?\<8 ,h[ d y multiply ;e 0xf an
imag9>y numb] ( "ts8 ,h[ c ! claim t
e^i.? .k cos .?+isin .? possibly 2 a
!orem & n1 z ,i ?"\ 9 hi< s*ool1 "s sort
( cock-ey$ def9i;n8
  ,well1 n[ t we h =mulas =! trig &
expon5tial func;ns 9 t]ms ( ^! 9f9ite
s]ies1 we cd apply ^? =mulas 6all possi#
values ( ;x, ev5 if ;x is -plex4 ,9
"picul>1 multiply+ ?+s \ & !n gr\p+ !
r1l & imag9>y t]ms gives
  e^i.?
    .k #1+?(i.?)_/1&#+?(i.?)^2"_/2&#+
      ?(i.?)^3"_/3&#+?(i.?)^4"_/4&#+
      ?(i.?)^5"_/5&#+?(i.?)^6"_/6&#+
      ?(i.?)^7"_/7&#+ '''
                                      #g
    .k #1+i?.?_/1&#-?.?^2"_/2&#-i?.?^3"
      _/3&#+?.?^4"_/4&#+i?.?^5"_/5&#-
      ?.?^6"_/6&#-i?.?^7"_/7&#+ '''
    .k @(1-?.?^2"_/2&#+?.?^4"_/4&#-
      ?.?^6"_/6&#+ '''@)+i@(?.?_/1&#-
      ?.?^3"_/3&#+?.?^5"_/5&#-?.?^7"_/
      7&#+ '''@)
    .k cos .?+isin .?_4
,s ,eul]'s ,!orem re,y is "s sort (a
!orem1 at l1/ if we c "sh[ ju/ify all !
/eps 9 ? plausi# b h>dly prov5 calcul,n4
  ,? is only ! t9ie/ 9troduc;n 6! id1s (
9f9ite s]ies---a major "p ( ma!matics
"?\t hi/ory---& 6! idea ( do+ calculus
on func;ns (! -plex numb]s4 ,? latt]
idea gives rise 6a/.d+ly b1uti;l
results1 &a gd case c 2 made t :at !
#19th c5tury 0 ab (at l1/1 9 ma!matics)
0 le>n+ h[ 6d all ! ?+s we d 9 calculus
6! new func;ns (a -plex v>ia#4 ,!
result+ !ory has deep & lovely implic,ns
9 "s (! mo/ unexpect$ places4 ,= 9/.e1 x
turns \ t ! be/ e/imates (! 4tribu;n (
prime numb]s >ises f ! /udy (! z]os (!
                                      #h
,riemann zeta func;n on ! -plex plane1
h>dly ! sort ( ?+ "o wd h expect$4
  ,9 %ort1 d -e back = m4 ,x only gets
bett]---deep] & m b1uti;l---f "h4




















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