Timeline for The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2015 at 15:10 | vote | accept | usul | ||
| Apr 5, 2015 at 9:02 | comment | added | usul | Thanks for the response. Stepping back, I guess the takeaway is that the question is probably not very interesting. If $f(x) = o(x)$, then the $1 +$ is essentially noise in the growth rate (right?), and you're looking at $\sim \left(\frac{x}{f(x)}\right)^{f(x)}$. So the cases $f(x) = \omega(x)$ are more interesting, but they're not that interesting either because they're all something like $e^{\Theta(x)}$. | |
| Apr 5, 2015 at 7:19 | comment | added | Steven Stadnicki | The first of these can be attained much more easily: $(1+\sqrt{x})^{\sqrt{x}} = \sqrt{x}^{\sqrt{x}}\cdot(1+\frac1{\sqrt{x}})^{\sqrt{x}}$, and the latter clearly goes to $e$ as $x\to\infty$. | |
| Apr 4, 2015 at 21:09 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
edited body
|
| Apr 4, 2015 at 19:41 | comment | added | Joe Silverman | @Turbo In this context, $\sim$ means "asymptotic to". So $F(x)\sim G(x)$ is a shorthand notation for $\lim F(x)/G(x)=1$, or alternatively $F(x) = G(x)(1+o(1))$. | |
| Apr 4, 2015 at 18:54 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 494 characters in body
|
| Apr 4, 2015 at 18:48 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 494 characters in body
|
| Apr 4, 2015 at 18:40 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 494 characters in body
|
| Apr 4, 2015 at 18:28 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 494 characters in body
|
| Apr 4, 2015 at 18:19 | comment | added | Geoffrey Irving | Do you mean $e^x (1 + o(1))$? | |
| Apr 4, 2015 at 18:17 | comment | added | Turbo | what is the $\sim$ difference? | |
| Apr 4, 2015 at 18:12 | history | answered | Gerald Edgar | CC BY-SA 3.0 |