sephiseraph
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Howdy Ho, partner.
I have a series of functions [tex]{f_{n}}[/tex] with [tex]f_{n}(x) := x^{n} / (1 + x^{n})[/tex] and I am investigating the pointwise limit of the sequence [tex]f_{n}[/tex] over [0, 1] to see if it converges uniformly.
I found the pointwise limit f(x) to be [tex]f(x) = lim_{n\rightarrow\infty} x^{n} / (1 + x^{n})[/tex], f(x) = 0 for x in [0, 1), f(x) = 1/2 for x = 1.
My problem here comes in finding out if the sequence converges uniformly or not. Intuition tells me that, since f(x) is not continuous, there cannot be uniform convergence on [0, 1]. However we also have
[tex]sup\left\{|f_{n}(x) - f(x)| : x \in \left[0, 1\right]\right\} = sup\left\{x^{n} / (1 + x^{n}) : x \in \left[0, 1\right)\right\}[/tex] which tends to 0 as n tends to infinity, which is a sufficient condition for uniform convergence. Where have I gone wrong here or how do I make sense of this?
I have a series of functions [tex]{f_{n}}[/tex] with [tex]f_{n}(x) := x^{n} / (1 + x^{n})[/tex] and I am investigating the pointwise limit of the sequence [tex]f_{n}[/tex] over [0, 1] to see if it converges uniformly.
I found the pointwise limit f(x) to be [tex]f(x) = lim_{n\rightarrow\infty} x^{n} / (1 + x^{n})[/tex], f(x) = 0 for x in [0, 1), f(x) = 1/2 for x = 1.
My problem here comes in finding out if the sequence converges uniformly or not. Intuition tells me that, since f(x) is not continuous, there cannot be uniform convergence on [0, 1]. However we also have
[tex]sup\left\{|f_{n}(x) - f(x)| : x \in \left[0, 1\right]\right\} = sup\left\{x^{n} / (1 + x^{n}) : x \in \left[0, 1\right)\right\}[/tex] which tends to 0 as n tends to infinity, which is a sufficient condition for uniform convergence. Where have I gone wrong here or how do I make sense of this?