WAp to print the sum of the series 1-x/2
+ x²181 *341....xn/(n+1) -
exponential series,
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The Number e and the Exponential Function
Michael Fowler
Disclaimer: these notes are not mathematically rigorous. Instead, they present quick, and, I hope, plausible, derivations of the properties of e, ex and the natural logarithm.
The Limit limn→∞(1+1n)n=e
Consider the following series: (1+1), (1+12)2, (1+13)3, ..., (1+1n)n,... where n runs through the positive integers. What happens as n gets very large?
It’s easy to find out with a calculator using 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+1n)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+1n)n goes to a definite limit. It can be proved mathematically that (1+1n)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+1n)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:
(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+...+xn
To use this result to find (1+1n)n, we obviously need to put x=1/n, giving:
(1+1n)n=1+n.1n+n(n−1)2!(1n)2+n(n−1)(n−2)3!(1n)3+...
.
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, n(n−1)/n2 tends to 1, and so does n(n−1)(n−2)/n3. . 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:
1+1+12!+13!+14!+...
And, the larger we take n, the more accurately the terms in the binomial series can be simplified in this way, so as n goes to infinity this simple series represents the limiting value of (1+1n)n . Therefore, e must be just the sum of this infinite series.
(Notice that we can see immediately from this series that e is less than 3, because 1/3! is less than 1/22, and 1/4! is less than 1/23, and so on, so the whole series adds up to less than 1 + 1 + ½ + 1/22 + 1/23 + 1/24 + … = 3.)