Question

7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power series.]

Answer 1

solution - first we will find right hand limit & left hand limit floth) - flo) = line [: flod=0 hot lim fro-h) -fo) = line the hot - O m ehr O right hand Limit = Left hand linit - limit exist Noco, lim [f(h)- f(o) . hsoh ht lim f(x) = lim fr) = Lim, ffx) = 0 diame = f is differentiable at o with f'(o)=0 Now for x70 so = 2 f(x) . xant 20-70 f"(x) = 2 (2- ) for) f"(x) = 4 **) fure) In general f)(x) = Pan ( 1 ) f(x) where Pan ( 1 ) is a polynomial of degree an in a Now , Let (x) is true for n ,then for ntl we have Antil(a) = (Pan (th) f(x) ) te Pan (B) Pers+ ron / f'(x) 1:46:3, frys = fh Panli) + barn (A)] flv ". The last step is a polynomial in a of degree 3h+

Nyo fel) (0) = Parma) (5) film) limitaw - limlen (5) 40) D - f is ole'ffe routjable of all ordless & fm(o)=0 ; V NEN set g (x) = p ) eta then it is clear that 9'(x) = 8 (1) e/42 & all desivatives off have this form as we seen earlier beet they all vainish as a tot & x o . f does not have a convergent power series expansion. g'(x) form as we xo .