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A proof for Gauss' test for convergence

  1. Apr 22, 2013 #1
    1. The problem statement, all variables and given/known data
    if [itex]\frac{a_n}{a_{n+1}}=1+\frac{p}{n}+β_n[/itex] and β converges absolutely, then at the infinity the sequence a is of the same order as [itex]\frac{c}{n^p}[/itex].

    2. Relevant equations
    Basic convergence test(Cauchy and d'Alembert's tests)

    3. The attempt at a solution
    I find this question quite interesting because it sunsequently proves Gauss' test for convergence without resorting to Raabe's test, but now has no idea where to begin. I try to write it as [itex]n(\frac{a_n}{a_{n+1}}-1)=p[/itex]but this basically returns to the Raabe's test. The question prior to this one asks to prove that if [itex]\frac{a_n}{a_{n+1}}=1+β_n[/itex], then a has a limit. But I cannot find any ways of applying it to this question. Thanks in advance.
  2. jcsd
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