Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20120302 06:37:49
Proof for definitions of e?What is the proof for the two common definitions of e? The continuous compounding and the sum over inverse factorials? #2 20120302 06:52:41
Re: Proof for definitions of e?hi Alex23,my duodecimalsystemloving friend The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #3 20120302 07:10:36
Re: Proof for definitions of e?
Hi Alex23. #4 20120302 07:15:48
Re: Proof for definitions of e?
Exactly! Also how does an exponent x of e transfer to the specific position in the definitions. How these connections where discovered? #5 20120302 07:17:24
Re: Proof for definitions of e?Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #6 20120302 08:10:44
Re: Proof for definitions of e?hi Alex23 and everyone, but that third term implies that the fourth term is and that implies the next term is ........ and so on. They quickly grasp that this series continues for ever. Then I switch to considering functions of the form We look at cases like a = 2, a= 3, a = 4, a = ½, a = 2 etc The graphs all have properties in common (such as the shape and they go through x=0, y=1) so we look at those properties and especially consider the gradient at any point and the gradient at x = zero. as this last limit is just the gradient at x = 0 and inspection of the family of graphs shows each clearly has some fixed gradient value at (0,1) for each 'a'. Now this family of curves can clearly have every possible gradient at x = 0 by choosing a suitable 'a'. So define e to be the value of a such that the gradient at zero is 1. then So this function is the one that has the property that it differentiates to give itself. It is then just a matter of setting x equal to 1 and you have the series expansion for e. Bob Last edited by bob bundy (20120302 08:37:39) You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #7 20120304 22:04:36
Re: Proof for definitions of e?Many thanks! #8 20120305 00:49:09
Re: Proof for definitions of e?Have a look at You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #9 20120305 01:48:36
Re: Proof for definitions of e?Hi bob The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #10 20120305 03:25:25
Re: Proof for definitions of e?hi Stefy, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #11 20120305 20:07:31
Re: Proof for definitions of e?Nice explanation.Think that from there it can be understood how e came to life in the first place. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #12 20120305 20:59:05
Re: Proof for definitions of e?You've told me about \cdot before. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #13 20120305 21:10:51
Re: Proof for definitions of e?Ok, if that's what you want. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #14 20120305 22:08:57
Re: Proof for definitions of e?Yes, you have and I do use it a bit ... depends what I'm doing. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #15 20120305 22:53:59
Re: Proof for definitions of e?I use out only when I have an error and don't know what it is. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment 