Timeline for Interpreting the Famous Five equation
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2010 at 16:38 | comment | added | Todd Trimble | The explanation given by Conway and Guy in The Book of Numbers is also in the spirit of these two answers. | |
| Jun 5, 2010 at 16:00 | comment | added | Andrey Rekalo | One may also interpret the limit as convergence of solutions to Euler's numerical scheme $$\frac{\phi_n-\phi_{n-1}}{\pi/n}=i\phi_{n-1},\quad \phi_0=1,$$ to a solution of the ODE $$\dot{\phi}=i\phi,\ t\in(0,\pi].$$ | |
| Jun 5, 2010 at 15:17 | comment | added | gowers | I agree. My answer to that is that it's fairly easy to prove that the function f(x) = lim (1 + x/n)^n has the property f(x+y) = f(x)f(y). | |
| Jun 5, 2010 at 13:35 | comment | added | Harald Hanche-Olsen |
I really do like these two answers, but I cannot help thinking that they should be accompanied by an equally intuitive explanation of why $\lim_{n\to\infty}(1+x/n)^n=e^x$. Which, in turn, boils down to a question of understanding the exponential function in the first place. I am not saying this is hard to achieve, but it is something I rarely see done well.
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| Jun 5, 2010 at 9:22 | history | answered | gowers | CC BY-SA 2.5 |