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I have a problem with a power series:

$$\ \sum_{n=1}^\infty x^ne^{-xn} ~~~~ x\in (0, \infty)$$

Could anyone explain how to check uniform, almost uniform and pointwise convergence? Not only proving convergence is problematic for me, but also exponential function makes me confused.

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  • $\begingroup$ What does almost uniform mean? What is the domain of your series? $\endgroup$
    – José Carlos Santos
    Apr 29, 2020 at 17:11
  • $\begingroup$ How would you calculate sum like this : $\sum (f(x))^n$ and what is the radius of convergence? $\endgroup$
    – openspace
    Apr 29, 2020 at 17:12
  • $\begingroup$ almost uniform: uniformly converge on intervals. I don't know how to translate it from my language :) $\endgroup$
    – Question
    Apr 29, 2020 at 18:16

1 Answer 1

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Hint : Let's consider $\displaystyle \sum_{n\le m} \left(\frac{x}{e^{x}}\right)^{n} = \frac{e^x - x(xe^{-x})^m}{e^x - x}$.

Then your series is limit(with respect to $m$) of following partial sum. When does it converge? When does the limit exist?

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  • $\begingroup$ How do you know the sum (right side)? For the the left, I would use Cauchy-Hadamard. The limit would exist for $\ 1 \leq x $ Am I right? $\endgroup$
    – Question
    Apr 29, 2020 at 18:23
  • $\begingroup$ That's simple progression type series. Consider three cases : $|x e^{-x}| < 1$ , $|xe^{-x}| =1$ and $|xe^{-x}| > 1$. $\endgroup$
    – openspace
    Apr 29, 2020 at 18:43
  • $\begingroup$ How do I know I have to consider especially these cases? Was my reasoning in the proevious post correct? $\endgroup$
    – Question
    Apr 29, 2020 at 18:47
  • $\begingroup$ @Question 1) there JUST three cases $|a| < 1$ , $|a|= 1$ or $|a| > 1$. 2) Look at this limit $\lim_{m \to \infty} \dfrac{e^x - x(e^{-x} x)^m}{e^x - x}$ when does it exist? 3) No you're wrong , you should also consider a term $e^{-x}$ $\endgroup$
    – openspace
    Apr 29, 2020 at 18:51
  • $\begingroup$ The limit exists for $\ m \leq 0 $ I don't know how can I find the idea of this limit? What do I should pay attention to? It isn't obvious for me $\endgroup$
    – Question
    Apr 29, 2020 at 18:57

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