Explore BrainMass

Explore BrainMass

    Taylor series, the convergence of sequence, and series of functions

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Let ƒn : R2  R be ƒn (x,y) = (x+y) / n
    a. Show that ƒn  0 , but convergence is not uniform.
    b. Show that the convergence ƒn  0 is uniform on bounded sets.

    © BrainMass Inc. brainmass.com February 24, 2021, 2:20 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/convergence-sequence-series-functions-15063

    Attachments

    Solution Preview

    Please see the attachment.

    1. Let ƒn : R2  R be ƒn (x,y) = (x+y) / n
    a. Show that ƒn  0 , but convergence is not uniform.
    Proof. Given , for any we know that there exists when , we have
    .
    So, , i.e., ƒn  0.

    But from the proof, we know that ...

    Solution Summary

    There are a variety of problems here, that relate to Taylor series, the convergence of sequence, and series of functions

    $2.19

    ADVERTISEMENT