Rational sums of inverse powers of fixed points of e

[jaydfox]

\(
\begin{eqnarray}
F(z) & = & \frac{-1}{c_k}\ln\left(\frac{e^{c_k-w}-c_k+w}{e^{c_k}}\right) {\HUGE |}_{w \to 0} \\
& = & \frac{-1}{c_k}\ln\left(\frac{c_k-(c_k-1)w+\mathcal{O}(w^2)-c_k+w}{c_k}\right) {\HUGE |}_{w \to 0} \\
& = & -\frac{\ln\left(\frac{\left(1-(c_k-1)\right)w}{c_k}\right)}{\ln\left(c_k\right)} {\HUGE |}_{w \to 0} \\
& = & -\log_{c_k}\left(\frac{c_k}{c_k}w\right) {\HUGE |}_{w \to 0} \\
& = & -\log_{c_k}(w) {\HUGE |}_{w \to 0} \\
\end{eqnarray}
\)

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.