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Bois-Reymond criterion for series

  1. May 5, 2014 #1
    The problem statement, all variables and given/known data.

    Prove that if ##\sum_{n=1}^{\infty} (a_n-a_{n-1})## converges absolutely and ##\sum_{n=1}^{\infty} z_n## converges, then ##\sum_{n=1}^{\infty} a_nz_n## converges.

    The attempt at a solution.

    I know that if ##Z_N=z_0+z_1+...+z_N##, then ##\sum_{n=0}^N a_nz_n= a_NZ_N-\sum_{n=0}^{N-1} Z_n(a_{n+1}-a_n)##

    I am not so sure how can I use the hypothesis given to this new expression or if it would be more convenient to express the original series in another way.
     
    Last edited: May 5, 2014
  2. jcsd
  3. May 5, 2014 #2
    Your formula ##\sum_{n=0}^N a_nz_n= a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)## is incorrect.

    ##a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)=0##.
     
  4. May 5, 2014 #3
    I've corrected it
     
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