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    Proving the Root Test

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    A) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity"

    ps. {An} is a sequence of non-zero complex numbers

    b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle.

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    https://brainmass.com/math/real-analysis/limits-proving-root-test-assuming-ratio-test-97019

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    Problem:

    a) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity"

    ps. {An} is a sequence of non-zero complex numbers

    b) Prove that although the following power series have R=1
    sum(nz^n) does not converge on any point of the unit circle

    Solution:

    a) First off all, I have to make a correction to the above statement:
    it is not sqrt|An| under the limit, but |An|^(1/n) or in other words, it is not "square" root, but root of order "n" (or fractional power).
    Now, we can rephrase the theorem which we ...

    Solution Summary

    The root test is proven assuming ratio test.

    $2.49

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