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Finding convergence of this series using Integral/Comparison

  1. Apr 13, 2017 #1
    1. The problem statement, all variables and given/known data
    series from n = 1 to infinity, (ne^(-n))

    2. Relevant equations


    3. The attempt at a solution
    I want to use integral test.
    I know this function is:
    positive (on interval 1 to infinity)
    continous
    and finding derivative of f(x) = xe^(-x) I found it to be ultimately decreasing.

    So integral test is applicable.

    I set up integral from 1 to infinity (xe^(-x))

    u = x du = dx
    v = -e^(-x) dv = e^(-x)

    -xe^(-x) + integral e^(-x)

    -xe^(-x) - e^(-x) = -e^(-x) (x + 1)

    evaluating from 1 to t

    [itex] -(1/e^t) (t+1) + 2/e^(1) [/itex]

    but now when I do lim t -> infinity, -(1/e^t) (t+1) should = infinity/infinity, which would mean [itex] a_n [/itex] would be divergent, but it is convergent.

    does anyone know where my mistake is??
     
  2. jcsd
  3. Apr 13, 2017 #2

    Mark44

    Staff: Mentor

    I get something slightly different from what you got, namely ##te^{-t} - e^{-t} + 2e^{-1} = (t - 1)e^{-t} + 2e^{-1}##
    But the fact that you're getting the indeterminate form ##[\frac \infty \infty]## doesn't tell you anything. You can use L'Hopital's Rule to actually evaluate your limit, which results in an actual value for this limit.
     
  4. Apr 13, 2017 #3

    Mark44

    Staff: Mentor

    BTW, this thread was marked as "Solved" but that didn't appear to really be the case, so I have changed it to "Unsolved."
     
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