Eudoxus (408 B.C to 355 B.C) did not have a spreadsheet, but he investigated the sequence defined by
and consider the ratio of
Here is spreadsheet output for the first 15 values in these sequences.
| 1 | 1 | 1 |
| 2 | 3 | 1.5 |
| 5 | 7 | 1.4 |
| 12 | 17 | 1.41666666666667 |
| 29 | 41 | 1.41379310344828 |
| 70 | 99 | 1.41428571428571 |
| 169 | 239 | 1.41420118343195 |
| 408 | 577 | 1.41421568627451 |
| 985 | 1393 | 1.41421319796954 |
| 2378 | 3363 | 1.41421362489487 |
| 5741 | 8119 | 1.41421355164605 |
| 13860 | 19601 | 1.41421356421356 |
| 33461 | 47321 | 1.41421356205732 |
| 80782 | 114243 | 1.41421356242727 |
| 195025 | 275807 | 1.4142135623638 |
Prove: The ratio
has a limit as n gets large, and show that that limit is
.
Make a spreadsheet to generate 20 to 30 terms of the sequences.
1. What happens when a(0) and b(0) are values other than 1?
2. Will it work for negative values?