Problem with inequality and number eProve that, for every positive integer n, the following inequality holds:where { x } denotes the fractional part function applied to number xI can;t think of any solution.

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Problem with inequality and number eProve that, for every positive integer   n, the following inequality holds:where   {  x  } denotes the fractional part function applied to number   xI can;t think of any solution.

Falpo359 · Accepted Answer

Because  e  =  2  +      1          2      !        +      1          3      !        +  .  .  .  +      1          n      !        +  .  .  .  =  =  2  +      1          2      !        +      1          3      !        +  .  .  .  +      1          n      !        +      1          (      n      +      1      )      !            (    1    +          1              n        +        2              +          1              (        n        +        2        )        (        n        +        3        )              +    .    .    .    )    &amp;lt;  =  2  +      1          2      !        +      1          3      !        +  .  .  .  +      1          n      !        +      1          (      n      +      1      )      !            (    1    +          1              n        +        1              +          1              (        n        +        1                  )          2                      +    .    .    .    )    =  =  2  +      1          2      !        +      1          3      !        +  .  .  .  +      1          n      !        +      1          (      n      +      1      )      !        ⋅      1          1      −              1                  n          +          1                      =  =  2  +      1          2      !        +      1          3      !        +  .  .  .  +      1          n      !        +      1          n      ⋅      n      !        .Thus,   {  e  n  !  }  &amp;lt;      1    n   and we are done!

Problem with inequality and number e Prove that, for every positive integer n, the following inequality holds: n{n! xx e}<1,where {x} denotes the fractional part function applied to number x. I can;t think of any solution.

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