ExponentialFunction[1] - The Number e and the Exponential...

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previous index next PDF The Number e and the Exponential Function Michael Fowler, UVa Disclaimer: these notes are not mathematically rigorous. Instead, they present quick, and, I hope, plausible, derivations of the properties of e, e x and the natural logarithm. The Limit Consider the following series: where n runs through the positive integers. What happens as n gets very large? It’s easy to find out if you use a scientific calculator having the function x^y. The first three terms are 2, 2.25, 2.37. You can use your calculator to confirm that for n = 10, 100, 1000, 10,000, 100,000, 1,000,000 the values of are (rounding off) 2.59, 2.70, 2.717, 2.718, 2.71827, 2.718280. These calculations strongly suggest that as n goes up to infinity, goes to a definite limit. It can be proved mathematically that does go to a limit, and this limiting value is called e . The value of e is 2.7182818283… . To try to get a bit more insight into for large n , let us expand it using the binomial theorem. Recall that the binomial theorem gives all the terms in (1 + x ) n , as follows: To use this result to find , we obviously need to put x = 1/ n , giving: . We are particularly interested in what happens to this series when n gets very large, because that’s when we are approaching e . In that limit, tends to 1, and so does . So, for large enough n , we can ignore the n -dependence of these early terms in the series altogether! When we do that, the series becomes just: And, the larger we take n , the more accurately the terms in the binomial series can be simplified in this way, so as n
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This note was uploaded on 11/02/2009 for the course APMA 2102 taught by Professor Keyes during the Spring '08 term at Columbia.

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ExponentialFunction[1] - The Number e and the Exponential...

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