1 2 N ( 1 − 1 2 N ) t ≈ 1 2 N e − t 2 N what assumption are needed for this approximation to be useful.

Question

1          2      N                  (      1      −              1                  2          N                    )        t    ≈      1          2      N            e                            −          t                          2          N                    what assumption are needed for this approximation to be useful.

Brennan Frye · Accepted Answer

One way to define       e    x   is via      e    x    =      lim          n      →      ∞                  (      1      +              x        n            )        n  By re-arranging your forumula a bit, you get      1          2      N                  (      1      −              1                  2          N                    )        t    =      1          2      N                                                (            1            +                                                            −                  t                                                  2                  N                                            t                        )                    t                ⏟                            Compare to                               e            x                           for                   x          =                                    −              t                                      2              N                                            ≈      1          2      N            e          −                        −          t                          2          N                      ,at least for large enough   t.

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