Question #170270

Let F n(x) =1/n *e^(-n^2*x^2) if x element of R ,n=1,2 .....

Prove that fn tend to 0 uniformly on R ,that derivative of fn tend to 0 point wise on R ,but that the convergence of {fn'} is not uniform on any interval containing the origin


1
Expert's answer
2021-03-10T10:07:14-0500

Here 

limnfn(x)=0,x\underset{{n\rightarrow \infty}}{lim }f_n(x)=0, \forall x


And fn(x)=en2x2nf_n(x)=\dfrac{e^{-n^2x^2}}{n}


fn(x)=2xn2en2x2nf'_n(x)=\dfrac{-2xn^2 e^{-n^2x^2}}{n}


so fn(x)=2xnen2x2f'_n(x)=-2xne^{-n^2x^2}



Now 2xnen2x2-2xne^{-n^2x^2} attains maximum value 2e at x=1n\dfrac{2}{e} \text{ at } x=\dfrac{-1}{n}


 tending to 0 as n.n\rightarrow \infty.


Let us take the interval [a,b] containg 0.


Thus

  Mn=Supx[a,b]fn(x)f(x)M_n=\underset {x\in [a,b]}{Sup}|f_n(x)-f(x)|


  =Supx[a,b]2xnen2x2= \underset{x\in [a,b]}{Sup}|-2xne^{-n^2x^2}|


     =2e=\dfrac{2}{e} Which does not tends to zero as n.n\rightarrow \infty.


 Hence the sequence fn{f_n} is not uniformly convergent in any interval [a,b] containing the origin.  



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