Sorceress

mathematics

Feb 27th, 2015
397
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  1. Let,
  2. R = 1 + 2 + 3 + 4 + 5 + 6 + ...
  3. S = 1 - 1 + 1 - 1 + 1 - 1 + ...
  4. T = 1 - 2 + 3 - 4 + 5 - 6 + ...
  5.  
  6. Consider 1-S,
  7. 1-S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + ...)
  8. = 1 - 1 + 1 - 1 + 1 - 1 + ...
  9. = S
  10.  
  11. So S = 1/2
  12.  
  13. Also,
  14. 2T = 1 - 2 + 3 - 4 + 5 - 6 + ...
  15. + 1 - 2 + 3 - 4 + 5 + ...
  16. = 1 - 1 + 1 - 1 + 1 - 1 + ...
  17. = S = 1/2
  18.  
  19. So T = S/2 = 1/4
  20.  
  21. Next,
  22. R - T = 1 + 2 + 3 + 4 + 5 + 6 + ...
  23. - 1 + 2 - 3 + 4 - 5 + 6 + ...
  24. 0 + 4 + 0 + 8 + 0 + 12 + ...
  25. = 4 * ( 1 + 2 + 3 + 4 + ...)
  26. = 4 * R
  27. So,
  28. 3 * R = -T
  29. R = -T/3 = -(1/4)/3 = -1/12
  30.  
  31. Therefore,
  32. 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12
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