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Fast New Formulas for e

This page contains supplementary material referenced in the paper entitled "Improving the Convergence of Newton's Series Approximation for e," appearing in The College Mathematics Journal (Vol. 35, No. 1, Jan. 2004; pages 34-39), by Harlan J. Brothers.

Here is additional supplementary information.

For reference, this is Isaac Newton's series approximation for e, first published in 1669:

Newton's Direct method for approximating e

Below are new, rapidly converging series that are derived using the described methods.

Series for e:

New formula for e #1 (H.J. Brothers, 2004)
New formula for e #2 (H.J. Brothers, 2004)
New formula for e #3 (H.J. Brothers, 2004)
New formula for e #4 (H.J. Brothers, 2004)
New formula for e #5 (H.J. Brothers, 2004)
New formula for e  #6 (H.J. Brothers, 2004)
New formula for e #7 (H.J. Brothers, 2004)
New formula for e #8 (H.J. Brothers, 2004)
New formula for e #9 (H.J. Brothers, 2004)
New formula for e #10 (H.J. Brothers, 2004)
New formula for e #11 (H.J. Brothers, 2004)
New formula for e #12 (H.J. Brothers, 2004)
New formula for e #13 (H.J. Brothers, 2004)
New formula for e #14 (H.J. Brothers, 2004)
New formula for e #15 (H.J. Brothers, 2004)
New formula for e #16 (H.J. Brothers, 2004)
New formula for e #17 (H.J. Brothers, 2004)

Series for 1/e:

New formula for e #18 (H.J. Brothers, 2004)
New formula for e #19 (H.J. Brothers, 2004)
Formula for e #20 (H.J. Brothers, 2004)
New formula for e #21 (H.J. Brothers, 2004)
NOTE: Formula (20) appears in Gradshteyn and Ryzhik's Table of Integrals, Series, and Products.

Series for e x:

New formula for e #22 (H.J. Brothers, 2004)
New formula for e #23 (H.J. Brothers, 2004)
{x R | x > 0}
New formula for e #24 (H.J. Brothers, 2004)
x R
New formula for e #25 (H.J. Brothers, 2004)
{x R | x > 0}
NOTE: In Formula (25), when x = 2n, extremely rapid convergence can be obtained by squaring the result n times.

Related series:

New formula for 1 (H.J. Brothers, 2004)
{n R | n > 0}
New formula for 1/n (H.J. Brothers, 2004)
{n R | n > 0}
NOTE: A version of Formula (26) with n=1, appears in Gradshteyn and Ryzhik's Table of Integrals, Series, and Products.