Created
December 6, 2009 22:08
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| Stel s = 1 + 2 + 3 + 4 + 5 + ... | |
| -3s = (1 - 4)s | |
| = (1 + 2 + 3 + 4 + 5 + ...) - 2(2 + 4 + 6 + 8 + 10 + ...) | |
| = 1 - 2 + 3 - 4 + 5 - ... [*1] | |
| = 1 - (2 - 3 + 4 - 5 + ...) | |
| = 1 - (1 - 2 + 3 - 4 + 5 - ...) - (1 - 1 + 1 - 1 + ...) | |
| Lemma: | |
| stel t = 1 - 1 + 1 - 1 + 1 - ... | |
| 1 - t = 1 - (1 - 1 + 1 - 1 + 1 - ...) | |
| = 1 - 1 + 1 - 1 + 1 - ... | |
| = t | |
| => t = 1 / 2 [*2] | |
| (vervolg) | |
| = 1 - (1 - 2 + 3 - 4 + 5 - ...) - (1 / 2) [wegens *2] | |
| = 1 + 3s - (1 / 2) [wegens *1] | |
| => -6s = 1 - (1 / 2) | |
| => s = -1 / 12 |
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