Timeline for What's an example of a transcendental power series?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 2, 2012 at 4:37 | comment | added | Todd Trimble | Another lovely answer I only saw today. | |
| Sep 2, 2012 at 0:57 | comment | added | Noam D. Elkies | I hadn't realized that "lacuna" is cognate with "lake" (and indeed "lagoon"). As for the mathematics, it's also a power-series version of a Liouville number: the partial sums approximate $\sum_n t^{n!}$ closer than would be possible for an algebraic power series. | |
| Apr 14, 2010 at 12:57 | comment | added | Gerald Edgar | German "See" = English "lake" anyway | |
| Apr 14, 2010 at 6:03 | comment | added | Gerry Myerson | @Mariano, some seas (e.g., the Sea of Marmara) are smaller than some lakes. | |
| Apr 14, 2010 at 4:44 | comment | added | Gerry Myerson | @jlk, more details: let $x$ be the series, let $n$ be a positive integer (the degree of the poynomial hypothetically satisfied by $x$), and consider $1, x, x^2, \dots, x^n$. For $r$ sufficiently large, the coefficient of $t^{r!n}$ will be one in $x^n$ but zero in the smaller powers of $x$. | |
| Apr 14, 2010 at 3:43 | comment | added | jlk | As with Qiaochu Yuan's answer, could you write out some more details of the "sea-of-zeros" argument? | |
| Apr 14, 2010 at 3:39 | comment | added | Mariano Suárez-Álvarez | Well, that is classically called a lacunary series (from lacus, lake)... I guess classical authors regarded factorial-sized holes as less-than-sea-wide :) | |
| Apr 14, 2010 at 3:24 | comment | added | François G. Dorais | +1: For the colorful terminology. | |
| Apr 14, 2010 at 3:22 | history | answered | Gerry Myerson | CC BY-SA 2.5 |