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Math Help - proof of raabe criteria.

  1. #1
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    proof of raabe criteria.

    does someone know how to prove raabe's criteria for sums convergence?

    i tried to prove it with a hint of the book but i got for example if we get from planetmath website, i got that mu is smaller than 1.

    because when the sum is convergent:
    (1-a_n+1/a_n)*n=1>=mu

    i had a hint from my book that i need to prove that {na_n+1} is monotonely decreasing i.e for every n na_n+1>(n-1)a_n and then i only need to prove that c_n=(n-1)a_n-na_n+1 is a convergent sum, which i did but i didn't get the ineality i needed.

    any help is appreciated.

    here's the link from planetmath:
    http://planetmath.org/encyclopedia/RaabesCriteria.html
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  2. #2
    Super Member Rebesques's Avatar
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    Check what this implies for {\rm limsup}\frac{a_{n+1}}{a_n} and use the ratio test.
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