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Finding the value of p so that the series converges

  1. Apr 17, 2009 #1
    find the possible values of p so that the following converges

    infinity
    [tex]\sum[/tex]1/(ln(n)*np)
    n=2

    what i thought of doing was integrating to find the values of p such that the integral will give me an answer not infinity.

    [tex]\int[/tex]dx/(ln(x)*xp) from 2-infnity

    i thought of using substitution as
    t=ln(x)
    x=et
    dt=dx/x

    [tex]\int[/tex]dt/(t*et(p-1)) from ln2-infinty


    but i have no idea how to continue from here, i think that if p<=1 then my integral will diverge since lim(t->inf) will not be 0, but i am not sure, i know that with a series if lim(n->inf) is not 0 then the series diverges, is this true for integrals as well,
    even so, if i know that p<=1 diverges, this does not automatically mean that anything else converges, since lim(n->inf)An=0 doesnt mean necessarily converges, how do i find the values for p where i KNOW the integral diverges??
     
  2. jcsd
  3. Apr 18, 2009 #2
    Your argument works for p <= 1.

    To do arguments for p > 1, use a comparison test.. Show that each term (n>2) is less then
    1/n^p.
     
  4. Apr 18, 2009 #3
    so do i say:
    for any n>2 my series is smaller than 1/n^p, and if p>1 1/n^p converge, therefore mine must also diverge, as for p<1 i already proved when i said lim(n->inf) not 0, so my series must diverge, therefore the series only converges when p>1,

    do you see a better way of solving this?
     
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