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  1. Newsgroups: sci.math
  2. Path: sparky!uunet!spool.mu.edu!torn!newshub.ccs.yorku.ca!newshub.ariel.yorku.ca!cs911301
  3. From: cs911301@ariel.yorku.ca (CHAK MAN TSUI)
  4. Subject: Re: HELP: limit
  5. Message-ID: <1993Jan12.200603.4389@newshub.ariel.yorku.ca>
  6. Originator: cs911301@m09.ariel.yorku.ca
  7. Keywords: limit, sum
  8. Sender: news@newshub.ariel.yorku.ca (USENET News System)
  9. Organization: York University, Toronto, Canada
  10. References:  <C0pAC9.CtH@news.cso.uiuc.edu>
  11. Distribution: york
  12. Date: Tue, 12 Jan 1993 20:06:03 GMT
  13. Lines: 56
  14.  
  15.  
  16.  
  17.  
  18. This problem is probably related to the study of Gamma Function.
  19. Euler's Def. for Gamma Fn:
  20.                     infin
  21.                    /       -t  z-1
  22.       Gamma(z) =  |       e   t    dt   ;  Re(z)>0.
  23.                  /
  24.                    0
  25.  
  26. However, for the incomplete Gamma Function:
  27.                        infin
  28.                      /      -t  n-1
  29.       GAMMA(n,x) =  |      e   t   dt        Re(n)>0
  30.                    /
  31.                      x
  32.  
  33. This can be shown to be: 
  34.                                 n-1
  35.                                 ---      i
  36.                            -x   \       x
  37.      GAMMA(n,x)  = (n-1)! e     /      ---
  38.                                 ---     i!
  39.                                 i=0
  40.  
  41.  
  42. Then,
  43.  
  44.                             k+q
  45.                             ---    i
  46.      GAMMA(k+q+1,k)    -k   \     k
  47.      -------------- = e     /    ---
  48.         (k+q)!              ---   i!
  49.                             i=0
  50.  
  51. and, what you want is...
  52.  
  53.     
  54.  
  55.        lim      Gamma(k+q+1,k)
  56.                ----------------  = Ans.
  57.      k->infin       (k+q)!
  58.  
  59.  
  60. ---------------------------------
  61.   From then on, I don't know how to proceed anymore...
  62.   You may pick up any book on special functions, it usually contains
  63. details about Gamma function...
  64.   I hope this will help...
  65.   
  66. ---------------------------------
  67.   Will you post solutions to the newsgroup, if you solves it, or
  68. you got someone's solution about it... thanks.
  69.  
  70.                  Jacques.
  71.