On Mon, 3 Mar 2008 21:08:59 +0000 (UTC), magidin@[EMAIL PROTECTED]
(Arturo Magidin) wrote:
>In article <dqlos3hcm39q3dbl8l37ddg8pldkakt4jf@[EMAIL PROTECTED]
>,
>vinhkan <markrrivet@[EMAIL PROTECTED]
> wrote:
>
> [...]
>
>>I can't thank you enough.
>
>You're very welcome.
>
>>In retrospect, one out of many
>>things that I missed was limit of the function 'e'.
>
>Careful. 'e' is a number, a constant. Sure, you can take a constant
>function whose value is always e, but we do not generally consider
>specific numbers to be "functions". So calling e a function is
>probably not what you want to say.
>
>Now, the constant 'e' has many definitions. One can define it to be
>
>limit (x->infinity) ( 1 + (1/x) )^x
>
>which is the amount you would have after 1 year if you deposited 1
>dollar at 100% interest, "compounded eveyr instant".
>
>An alternative way to define e is to look for a function f(x) that
>satisfies the following two conditions:
>
> f(0) = 1; and f'(x) = f(x).
>
>It turns out you can prove that this is an exponential function (a
>function of the form f(x) = a^x for some a>0, a<>1), and one defines
>the number "e" to be the base of this exponential. (This is how the
>number was first obtained, by Leonhard Euler, and it was named 'e' in
>honor of him).
>
>Or you can define the function ln(t) via integrals, letting ln(a) be
>the area under the graph of 1/x, over the line y=0, and between the
>lines x=1 and x=a (if a>1 you take the area, if 0 < a < 1 you take
>minus the are). Then you can ask "for which value of a is ln(a)=1"?
>And the answer is the number 'e'.
>
>In any case, whichever way you pick to define 'e', one of the
>im****tant properties is precisely that
>
>lim (n-->infinity) (1 + (1/n) )^n = e
>
>and you should learn that limit, because it shows up a lot (just as
>the number e, artificial as any of its definitions may seem to be,
>shows up in the most unexpected places).
That's very interesting. It seems as though the constant e is
masquerading as a constant and something else that I cannot find the
words for. Can you give me some examples of where the number e pops
up? I will look into on my own fore sure, but would just be interested
on your thoughts on the subject. Thanks again.
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