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  1. Newsgroups: sci.math
  2. Path: sparky!uunet!gatech!news.byu.edu!ux1!mica.inel.gov!guinness!opal.idbsu.edu!holmes
  3. From: holmes@opal.idbsu.edu (Randall Holmes)
  4. Subject: Re: What's going on here ???
  5. Message-ID: <1992Sep11.152945.2267@guinness.idbsu.edu>
  6. Sender: usenet@guinness.idbsu.edu (Usenet News mail)
  7. Nntp-Posting-Host: opal
  8. Organization: Boise State University Math Dept.
  9. References: <1992Sep11.214701.783@csc.canterbury.ac.nz>
  10. Date: Fri, 11 Sep 1992 15:29:45 GMT
  11. Lines: 76
  12.  
  13. In article <1992Sep11.214701.783@csc.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
  14. >The following rather silly problem cropped up in our department just now.
  15. >
  16. >Evaluate    S = 0! - 1! + 2! - 3! + 4! - ....
  17. >
  18. >(yes, I know it's divergent, but blame Euler, not me.)
  19.  
  20. It converges in the p-adic numbers, does it not?
  21.  
  22. >
  23. >------------------------
  24. >Crazy solution (1)
  25. >
  26. >    oo           oo     oo            oo     oo          oo
  27. >   \~~   n      \~~   n /  n -x       /  -x \~~  n       /   -x 1
  28. >S = > (-1) n!  = > (-1) | x e  dx  =  | e    >(-x) dx  = |  e  --- dx
  29. >   /__          /__    /             /      /__         /      1+x
  30. >   n=0          n=0    0             0      n=0         0
  31. >
  32. >  = .5963..
  33. >
  34. >-----------------------
  35. >Crazy solution (2)
  36. >
  37. >                                      2      3      4
  38. >S = f(1)   where  f(x) = 0! - x 1! + x 2! - x 3! + x 4! - ...
  39. >
  40. >                              2      3
  41. >So   (x f(x))' = 1! - x 2! + x 3! - x 4! + ...
  42. >
  43. >               = (1 - f(x))/x
  44. >        2                                                     1        1
  45. >Thus   x f' + (1+x)f = 1         this has integrating factor  - exp(- ---)
  46. >                                                              x        x
  47. >
  48. >Thus  [ x exp(-1/x) f(x) ]' = exp(-1/x)/x
  49. >
  50. >                               x
  51. >                              /
  52. >thus      x exp(-x) f(x)    = | exp(-1/t)/t dt      now put x=1
  53. >                              /
  54. >                             0
  55. >                               1
  56. >                              /
  57. >thus              f(1)/e  =   | exp(-1/t)/t dt      and put u=1/t
  58. >                              /
  59. >                             0
  60. >                               oo
  61. >                              /
  62. >                          = e | exp(-u)/u du    =  .5963..
  63. >                              /
  64. >                             1
  65. >
  66. >being essentially the same integral as in crazy solution (1).
  67. >-----------------------
  68. >
  69. >So what's going on here ????
  70. >
  71. >Two crazy solutions to a crazy problem giving the same answer.   Why ?
  72. >
  73. >--------------------------------------------------------------------------
  74. >            Bill Taylor              wft@math.canterbury.ac.nz
  75. >--------------------------------------------------------------------------
  76. > MATH: necessary consequences of arbitrary axioms about meaningless things.
  77. >--------------------------------------------------------------------------
  78. >
  79.  
  80. Maybe it has something to do with the fact noted above.  Algebraic
  81. manipulations of formal series often do give sensible results, which
  82. might _really_ work in an unintended interpretation.  After all,
  83. ...1111 (base 2) _is_ -1 in the 2-adics.
  84. -- 
  85. The opinions expressed        |     --Sincerely,
  86. above are not the "official"    |     M. Randall Holmes
  87. opinions of any person        |     Math. Dept., Boise State Univ.
  88. or institution.            |     holmes@opal.idbsu.edu
  89.