- #1
Rectifier
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The problem
I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##.
Relevant equations
##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1)
##f_n## converges uniformly if ## \lim_{n \rightarrow \infty} || f_n - f || = 0 \ \ \ \ \ ## (2)
The attempt
##f_n(x) = f(nx) =nxe^{-nx} ##
Pointwise? (1)
## f(x) = \ \lim_{n \rightarrow \infty} f_n(x) = \lim_{n \rightarrow \infty} nxe^{-nx} = \lim_{n \rightarrow \infty} \frac{nx}{e^{nx}} = 0##
Uniformly? (2)
## 0 = \ \lim_{n \rightarrow \infty} || f_n - f || = \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || ##:I am not sure how to continue from here and wether the last step was correct:
I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##.
Relevant equations
##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1)
##f_n## converges uniformly if ## \lim_{n \rightarrow \infty} || f_n - f || = 0 \ \ \ \ \ ## (2)
The attempt
##f_n(x) = f(nx) =nxe^{-nx} ##
Pointwise? (1)
## f(x) = \ \lim_{n \rightarrow \infty} f_n(x) = \lim_{n \rightarrow \infty} nxe^{-nx} = \lim_{n \rightarrow \infty} \frac{nx}{e^{nx}} = 0##
Uniformly? (2)
## 0 = \ \lim_{n \rightarrow \infty} || f_n - f || = \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || ##:I am not sure how to continue from here and wether the last step was correct:
## \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || = 0 ##
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