# 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,

Here is additional supplementary information.

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

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

**Series for e**:

**Series for 1/e**:

NOTE: Formula (20) appears in Gradshteyn and Ryzhik's Table of Integrals, Series, and Products. |

**Series for e ^{x}**:

{x ∈
R | x > 0} |

x ∈ R |

{x ∈
R | x > 0}NOTE: In Formula (25), when x = 2 ^{n}, extremely rapid convergence can be obtained by squaring the result n times. |

**Related series**:

{n ∈ R | n > 0} |

{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. |