Howdy Ho, partner.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Pointwise vs uniform convergence

**Physics Forums | Science Articles, Homework Help, Discussion**