Calculating the Value of e

There are several ways to calculate the value of e. Let's look at the historical development.

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Using a Binomial Expansion

If n is very large (approaches infinity) the value of MATH approaches e.

The largest that Scientific Notebook can handle is about n = 100,000 and this is only correct to the 4th decimal place.

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Another Expansion

As n becomes very small, MATH approaches the value of e.

We can obtain reasonable accuracy with a very small value of n.

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The graph of y = (1 + n)1/n is as follows:
calculating_e__14a.png

(There is actually a "hole" at n = 0. Can you understand why?)

Newton's Series Expansion for e

The series expansion for e is MATH

Replacing x with 1, we have:

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We can write this as:

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This series converges to give us the answer correct to 9 decimal places using 12 steps:

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Brothers' Formulae

Recently, new formulae have been developed by Brothers (2004) which make the calculation of e very efficient.

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We only need 6 steps for 9 decimal place accuracy:

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Graphical Demonstration of e

The area under the curve $y=\frac{1}{x}$ between 1 and e is equal to 1 unit2.
calculating_e__33.png

Reference:

Brothers, H.J. 2004. Improving the convergence of Newton's series approximation for e. College Mathematics Journal 35(January):34-39. Available at http://www.brotherstechnology.com/docs/icnsae_(cmj0104-300dpi).pdf.

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