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Consider the series generated by Harlan J. Brothers's method for the number $e$

Can they be improved again and again or is there a limit so there exists a BEST approximation to $e$?

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The paper is available on Harlan J. Brothers's homepage. – Martin Jun 14 '13 at 14:23

It depends on what you mean by "improved" and "BEST." The paper @Martin referenced discusses "an algebraic assessment based on the decimal place accuracy (d.p.a.) that is gained from a given number of terms; the greater the d.p.a., the faster the series converges." This is different from the run-time cost of the computation for which a smaller number of processor operations indicates that the algorithm is inherently more efficient.

While the combination of T terms results in new "compressed" terms containing a polynomial of degree T-1, the algorithm appears reasonably linear (to at least T=100). Computing 30,000 terms with Newton's method achieves 121,287 correct digits. Obtaining the same accuracy by computing 3,000 terms with t=10 requires very close to 1/10 the time - it's 9.9 times as fast (these computations are done in Mathematica 9). Computing just 300 terms with T=100 is 96 times as fast. However, for T=1000, things slow down - 30 terms are only 705 times as fast. To my knowledge, there has been no evaluation of the true computational efficiency of the approach.

Personally, I find value in the aesthetic appeal of expressions resulting from combinations that use low values of T. Here is a more comprehensive list of approximations that supplements the paper:

In short, Newton's series can indeed be algebraically improved again and again. The extent to which such "improvements" can continue to be of practical value is a different matter. For proven efficiency, you can reference the work of the computational wizards cited here:

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