code class="asciimath">Let {a_(n)} be a sequence of positive numbers and \sigma be a positive number. (A)(3 points) If there exists some positive integer N such that (lnn)^(-1)*ln((1)/(a_(n)))>=1+\sigma holds for n>=N Prove that \sum_(n=1)^(\infty ) a_(n) is convergent. (B)(3 points) If there exists some positive integer N such that (lnn)^(-1)*ln((1)/(a_(n)))<=1 holds for n>=N Prove that \sum_(n=1)^(\infty ) a_(n) is divergent. (C)(4 points) By using the results in (A),(B) or otherwise, find all bin(0,\infty ) such that \sum_(n=1)^(\infty ) b^(ln(n^(3)+1)) is convergent.