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For what p does this converge?

  1. Feb 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Given the limit (B-->inf) [(lnB)^(-p+1)] / (-p+1) - [(ln1)^(-p+1)] / (-p+1)

    For what p does this converge?

    3. The attempt at a solution

    For the left side of the minus sign,
    if 1-p<0 --> 0
    if 1-p>0 --> inf

    but for the ln(1) on the right side of the minus sign, ln(1)=0, so it would be
    if 1-p<0 then you would get a zero in the denominator (since ln(1) would be raised to a negative power) and therefore ---> inf
    if 1-p>0 then you would have zero as the numerator, but be left with infinity as the other value and hence ---> inf

    The book says that for p>1 then it will converge. But how can that be, if you'll end up with a negative exponent for the 0??
    Last edited: Feb 16, 2008
  2. jcsd
  3. Feb 16, 2008 #2


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    Staff Emeritus
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    Is that really "ln(1)" in the last term? ln(1)= 0 of course. Also parentheses would help.
    Finally, you should not use "B" and "b" to mean the same thing but since there is no "b" in function I am going to assume you are. I am also going to assume that is [ln(b^(-p+1))]/[-p+1] - [(ln1)^(-p+1)] / [-p+1] which, since ln(1)= 0 is the same as ln(B-p+1/(-p+1)= (-p+1)ln(B)/(-p+1)= ln(b). So you have [itex]\lim{b\rightarrow \infty} ln(b) which does not converge for any p.

    If you meant something else please clarify.

    That converges to ln(B
  4. Feb 16, 2008 #3
    Please do not assume that when I stated (lnB)^(-p+1) I meant ln(B^-p+1) because I did not. The exponent is for the entire natural log expression. Therefore your statement regarding the answer to be just lnB is invalid. Thank you though.
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