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 - The Fusion of Science and Community**

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

# Pointwise vs uniform convergence

Loading...

Similar Threads for Pointwise uniform convergence | Date |
---|---|

Uniform vs pointwise convergence | Apr 21, 2011 |

Pointwise vs. uniform convergence | Jul 29, 2010 |

Pointwise Limits and Uniform Convergence Help | Apr 23, 2010 |

Uniform and Pointwise Convergance of Functions | Apr 18, 2009 |

Question about pointwise convergence vs. uniform convergence | Mar 13, 2009 |

**Physics Forums - The Fusion of Science and Community**