Fast New Formulas for e
This page contains supplementary material for the paper entitled "Improving the convergence of Newton's series approximation for e," appearing in The College Mathematics Journal (Vol. 35, No. 1,
For reference, here is Isaac Newton's series approximation for e, first published in1669:
Below are new, rapidly converging series that are derived using the described methods.
Series for e:
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Series for 1/e:
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| NOTE: Formula (20) appears in Gradshteyn and Ryzhik's Table of Integrals, Series, and Products. |
Series for e x:
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{x Î R | x ¹ 0}
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x Î R |
{x Î R | x ¹ 0}
NOTE: In Formula (25), when x = 2n, extremely rapid convergence can be obtained by squaring the result n times. |
Related series:
{n Î R | x ³ 0} |
{n Î R | x > 0} |
NOTE: A version of Formula (26) with n=1, appears in Gradshteyn and Ryzhik's Table of Integrals, Series, and Products. |