jimlucky的部落格

For a arbitrary function f....
Want use a Polynomial to calculate f
If f is n-times diff at a
Consider a poly P(x) where
Pn(x)=C0+C1(x-a)+..+Cn-1(x-a)^(n-1)
let f=p(x)
then we can find Cn=f^(n)(a)/n! easily！
so Pn(x)=f(a)+f`(a)(x-a)+f``(a)(x-a)^2+...
+f^(n-1)(a)/(n-1)!(x-a)^(n-1)
Pn(x) are called Taylor Poly

if n goes to infinite
we say Pn(x) is Taylor Series
so every smooth function have Taylor Series

we have this conclusion.

Taylor`s Thm
for any n-times diff function f
f(x)=Pn(x)+Rn(x) where
Rn(x)=f^(n)(c)/n!(x-a)^n
c is between x and a
and Rn(x) is called remainder

if Rn(x)-->0 as n goes to infinite
then we say f can write by Power Series.
and this Power series is Tayor Series
Now get f(x)=f(a)+f`(a)(x-a)+.....
so we can consider the radius of convergence R
define f is analytic at a
(R isn`t zero, otherwise Rn(x) isn`t zero )

Give a counterexample if f only smooth
f(x)=e^(1/x) when x isn`t zero
0 when x =0
then f is infinite-time diff at 0
f^(n)(0)/n!=0 for all n
so f(x)=0+0+.......+0+......？
but f(x) doesn`t zero function

What`s wrong？
Because Rn(x) isn`t zero ,for all n
even though
0+0+....+0+....has convergence radius
and it`s infinite
but also can`t hold