Exponential_Function

Exponential_Function - previous index next The Number e and...

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previous index next 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 1 lim(1 ) n n n e →∞ += Consider the following series: 23 11 1 (1 1), (1+ ) , (1 ) , . .., (1 ) ,. .. n n ++ + 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 1 (1 ) n n + 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, 1 ) n n + goes to a definite limit. It can be proved mathematically that 1 ) n n + 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 1 ) n n + 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: ) ) (2 ) ) 1 ... 2! 3! n n nn n x nx x x x −− + =+ + + + + To use this result to find 1 ) n n + , we obviously need to put x = 1/ n , giving: 1 1 ) ) ) ) 1 . ( ) ... 3! n n n n n + =+ + + + . 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, ( ) 2 1/ n tends to 1, and so does .
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Exponential_Function - previous index next The Number e and...

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