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Pick any test to determine convergence

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Use any method or test to see if the series converges or diverges.


    2. Relevant equations

    The series:
    ((1/n) - (1/n^2))^n

    3. The attempt at a solution
    Well the integral test won't work because there's no real integral for that according to Wolfram Alpha. Also if you try the limit comparison test, the first thing is to determine where the function goes if n goes to ∞, but when it does, you get something like (0)^∞. What other test can I use?

    Thanks.
     
  2. jcsd
  3. Nov 14, 2012 #2

    LCKurtz

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    Actually, the series would be the sum of those terms. I would start by writing it as$$
    a_n =\left(\frac {n-1}{n^2}\right)^n$$So, thinking intuitively, for large ##n##, that ##-1## in the numerator isn't going to matte much so ##a_n## is sort of like$$
    \left(\frac {1}{n}\right)^n$$Does that give you any comparison test ideas?
     
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