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.

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    $\begingroup$ The functions of your image aren't $n x e^{−x}$. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 16:39
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    $\begingroup$ 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

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

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  • $\begingroup$ 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

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