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 Rational sums of inverse powers of fixed points of e jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 11/21/2007, 08:14 AM jaydfox Wrote:$ \begin{eqnarray} F(z) & = & \frac{-1}{c_k}\ln\left(\frac{e^{c_k-w}-c_k+w}{e^{c_k}}\right) {\HUGE |}_{w \to 0} \\ & = & \frac{-1}{c_k}\ln\left(\frac{c_k-(c_k-1)w+\mathcal{O}(w^2)-c_k+w}{c_k}\right) {\HUGE |}_{w \to 0} \\ & = & -\frac{\ln\left(\frac{\left(1-(c_k-1)\right)w}{c_k}\right)}{\ln\left(c_k\right)} {\HUGE |}_{w \to 0} \\ & = & -\log_{c_k}\left(\frac{c_k}{c_k}w\right) {\HUGE |}_{w \to 0} \\ & = & -\log_{c_k}(w) {\HUGE |}_{w \to 0} \\ \end{eqnarray}$ Argh, I made a math error. Two, in fact. The first is that I subtracted -1 from 1 and got 0, when I should have gotten 2: $\left(1-(c_k-1)\right)w \ne c_k w$ Second, I incorrectly expanded an exponentiation. I didn't have paper and pencil handy to calculate, so I used Excel to get the answer by approximation. But I was subtracting the wrong set of numbers and got the wrong coefficient: $e^{c_k-w} \ne c_k-(c_k-1)w+\mathcal{O}(w^2)$ Okay, so, just to be clear, if w is a small number going to 0, then: $ \begin{eqnarray} e^{c_k-w} & = & e^{c_k}e^{-w} \\ & = & c_k\left(1-w+\frac{w^2}{2}+\dots\right) \\ & = & c_k-c_k w+\mathcal{O}(w^2) \end{eqnarray}$ Fortunately, this doesn't affect my initial derivation too badly: $ \begin{eqnarray} F(z) & = & \frac{-1}{c_k}\ln\left(\frac{e^{c_k-w}-c_k+w}{e^{c_k}}\right) {\HUGE |}_{w \to 0} \\ & = & \frac{-1}{c_k}\ln\left(\frac{c_k-c_k w+\mathcal{O}(w^2)-c_k+w}{c_k}\right) {\HUGE |}_{w \to 0} \\ & = & -\frac{\ln\left(\frac{\left(1-c_k\right)w}{c_k}\right)}{\ln\left(c_k\right)} {\HUGE |}_{w \to 0} \\ & = & -\log_{c_k}\left(\frac{1-c_k}{c_k}w\right) {\HUGE |}_{w \to 0} \\ & = & -\log_{c_k}(w)-\log_{c_k}\left(\frac{1-c_k}{c_k}\right) {\HUGE |}_{w \to 0} \\ \end{eqnarray}$ So I'm off by a constant factor, and I wasn't calculating the constant term anyway. So my initial result still holds. ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 07:55 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:11 PM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/20/2007, 08:14 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:31 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:43 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 09:31 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 01:15 AM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 08:14 AM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/21/2007, 08:50 AM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 09:35 AM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/21/2007, 12:52 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 06:20 PM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/22/2007, 06:25 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/22/2007, 08:32 PM

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