While using Euler's method for d y d x = y given y ( 0 ) = 1 I noticed that the approximate solution is ( 1 + 1 / s ) s x where s is the interval size. Plug in one for x and you get: e 1 = lim s → ∞ ( 1 + 1 / s ) s you get e. This has to be e because that is that is the solution to the differential equation. Is this a new proof that e = lim s → ∞ ( 1 + 1 / s ) s List item

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While using Euler&#039;s method for             d      y              d      x        =  y given   y  (  0  )  =  1 I noticed that the approximate solution is   (  1  +  1      /    s      )          s      x       where s is the interval size. Plug in one for   x and you get:      e          1        =      lim          s      →      ∞          (  1  +  1      /    s      )          s      you get   e. This has to be   e because that is that is the solution to the differential equation. Is this a new proof that  e  =      lim          s      →      ∞          (  1  +  1      /    s      )          s      List item

carponih7 · Accepted Answer

It depends on how you define   e. If you define e as                    e        :=                  lim                      s            →            ∞                                                (            1            +                          1              s                        )                    s                    Then (of course) it trivially follows that it is true (  e=  e end proof).Although if you define   e as the solution to the Initial Value Problem      {                                                      d              y                                      d              x                                                          =          y                                      y          (          0          )                                    =          1                        at the value   x  =  1, or if you define e some other way and prove that       e    x   is the solution to this IVP, and then in addition you prove (or are allowed to assume) that Euler&#039;s Method will in fact converge upon the unique solution to the IVP, then yes: what you have written would suffice as another proof that  e  =      lim          s      →      ∞                  (      1      +              1        s            )        s

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