0
$\begingroup$

I have a sequence of functions $f_{n}: [0,\infty) \to \mathbb{R}$ defined as $$f_{n}(x) = nxe^{-nx}.$$

Do i need to investigate both pointwise and uniform convergence of this sequence.

i draw first 5 functions here on this picture

So using this drawn functions I came to an suspicion that for $x\in [0,a)$ it converges towards $y=0$, and for $x\in(a,\infty)$ converges towards $x=0$.

I am not correct so i need a second opinion, also I am not sure what I should use for $a$.

So i will get that the sequence pointwise converges towards function $f$ but that function isn't continuous, so it will mean that this sequence doesn't converge uniformly.

Any help with choosing $a$ would help, also if I made any mistakes, I would really appreciate if you could point it out. Thank you in advance.

$\endgroup$
  • 1
    $\begingroup$ The functions of your image aren't $n x e^{−x}$. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 16:39
  • 1
    $\begingroup$ Please check that my edit is correct. I fixed the function to $nx e^{-nx}$ as these are the functions plotted. $\endgroup$ – Winther Jan 19 '16 at 16:40
  • $\begingroup$ Te has adelantado un segundo! $\endgroup$ – Julián Aguirre Jan 19 '16 at 16:40
1
$\begingroup$

Hints: $f_n\to 0$ pointwise (do the easy limit). For the uniform convergence, find the maximum of each $f_n$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ isn't there one interval that pointvise convergence is against y=0, as functions are getting really clsoe to y=0 $\endgroup$ – MathIsTheWayOfLife Jan 19 '16 at 16:45
  • $\begingroup$ @user246608, the interval of pointwise convergence is the whole $[0,\infty)$. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 16:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.