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Hard Problem concerning logarithms

  1. Nov 23, 2004 #1
    Hello all

    I need help with the problem attached below. I tried proof by induction, but cannot prove it. P(n) is a polynomial of degree 2n-2. I have to establish the recurrence relation.

    Any help is greatly appreciated!


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    Last edited: Nov 23, 2004
  2. jcsd
  3. Nov 23, 2004 #2


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    probably you are trying to prove too much, if all you want is to show that all derivatives of e^(-x^(-2)) vanish at zero, but anyway, notice that every time you take a derivative by the product rule, on one hand you differentiate the original function which multiplies that factor by x^(-3), so the degree of that factor goes down by 3. But when you differentiatiate the other factor, the one with negative powers of x, the pwoer of x goes down by 1.

    so when you clear denominators, you have to multiply by the larger negative power, which is x^(-3n) where n is the numbwer of tiomes you did this. On the other factor, when you multiply by this, since those powers only went down on e each time,l you get a polynomial in the top whiose degree goes up 2 each time, and it seems form looking at thes econd erivative that it was degree 2 that time, hence ingeneral is degree 2n-2.

    i.e. this is obvious from looking at the first two derivatives. I could be wrong of course.
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