I've been tinkering with Andrew's slog solution, since I think it probably has a much better claim to uniqueness for base e than my solution.

Anyway, I've only tested solutions for the 50-, 60-, 70-, and 80-term truncations. I've noticed that I can speed up the initial matrix solving by about 10-15% if I avoid the rationals created by dividing through by the factorials.

To be useful in a power series, I would have to divide those factorials back out anyway, so there's an additional (negligible) savings there as well.

Here are some sample solver times (matsolve in gp), in seconds. The rows in this chart are the number of rows/columns in the matrix, and the columns are with and without the rationals, as well as the approximate savings:

\( \begin{array}{c|cccc}

& Rationals && Integers & {\small Approx.}\ Savings\\

\hline

{\Large 50} & 5.066 {\small \pm 0.093} && 4.400 {\small \pm 0.063} & 0.666 \\

{\Large 60} & 17.325 {\small \pm 0.150} && 14.984 {\small \pm 0.415} & 2.341 \\

{\Large 70} & 48.661 {\small \pm 0.433} && 43.883 {\small \pm 0.387} & 4.778 \\

{\Large 80} & 123.464 {\small \pm 0.181} && 112.218 {\small \pm 0.474} & 11.245

\end{array}

\)

I don't know how much precision is lost if you try to solve for bases with non-rational logarithms. However, for a base b where ln(b) is rational, the solver times should be only slightly longer than for base e.

Anyway, I've only tested solutions for the 50-, 60-, 70-, and 80-term truncations. I've noticed that I can speed up the initial matrix solving by about 10-15% if I avoid the rationals created by dividing through by the factorials.

To be useful in a power series, I would have to divide those factorials back out anyway, so there's an additional (negligible) savings there as well.

Here are some sample solver times (matsolve in gp), in seconds. The rows in this chart are the number of rows/columns in the matrix, and the columns are with and without the rationals, as well as the approximate savings:

\( \begin{array}{c|cccc}

& Rationals && Integers & {\small Approx.}\ Savings\\

\hline

{\Large 50} & 5.066 {\small \pm 0.093} && 4.400 {\small \pm 0.063} & 0.666 \\

{\Large 60} & 17.325 {\small \pm 0.150} && 14.984 {\small \pm 0.415} & 2.341 \\

{\Large 70} & 48.661 {\small \pm 0.433} && 43.883 {\small \pm 0.387} & 4.778 \\

{\Large 80} & 123.464 {\small \pm 0.181} && 112.218 {\small \pm 0.474} & 11.245

\end{array}

\)

I don't know how much precision is lost if you try to solve for bases with non-rational logarithms. However, for a base b where ln(b) is rational, the solver times should be only slightly longer than for base e.

~ Jay Daniel Fox