Proof that lim(n-> inf) x^n/n!=0 by Stirling’s Approximation for Factorials (Part II) by waabu

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A Note is added to the original paper. An additional simpler proof of the limit is given.

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									                                                    xn
                                Proof that lim         =0
                                              n → ∞ n!
                  by Stirling’s Approximation for Factorials
                                    Part II

                                            ╬
                               Francis J. O’Brien, Jr., Ph.D.
                                Aquidneck Indian Council
                                       Newport, RI
                                                                      November 24, 2012



In reference to the original paper with same title, we add the following NOTE to page 4.

NOTE: An additional and simpler proof is as follows, using relation, n!≈ n n e − n 2πn ,
and the Product Rule for limits:
                                                    n
                                         ⎛ xe ⎞
                 n            n          ⎜ ⎟                   n
                                                        ⎛ xe ⎞
                                   = lim ⎝ ⎠ =
               x            x              n    1                   1
          lim      ≈ lim                            lim ⎜ ⎟ lim
                         n −n
         n → ∞ n! n → ∞ n e     2πn n → ∞ 2πn   2π n → ∞⎝ n ⎠ n → ∞ n

         =
           1 ⎛        xe ⎞
                           n
                                1
                                  =
                                    (0)n (0) = 0.
              ⎜ lim      ⎟ lim
           2π ⎝ n → ∞ n ⎠ n → ∞ n      2π




© Francis J. O’Brien, Jr., Aquidneck Indian Council <> All Rights Reserved

								
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