Sorry to answer my own question, but having thought about it more, I realize this is impossible unless you have arbitrarily high precision for all the 's. The reason is, as grows, the leading digits of will mainly be those of .
If one does have arbitrary precision, take . To compute this, increase until you have "enough" significant digits. Then to get , subtract from all the 's and repeat to get , and so on.
Obviously, your estimate of will be worse than that of , and in double precision I wasn't able to get more than 2 or 3 terms (depends on how large/small the ratios are).
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