Prove limit is zero A n = ( 1 / 3 ) ( 1 / 3 + 1 ) … ( 1 / 3 + ( n − 1 ) ) n !

Question

Prove limit is zero       A    n    =            (      1              /            3      )      (      1              /            3      +      1      )      …      (      1              /            3      +      (      n      −      1      )      )              n      !

Dustin Durham · Accepted Answer

One has  log  ⁡  (      A    n    )  =      ∑          k      =      0              n      −      1        log  ⁡      (                            1          3                +        k                    k        +        1              )    =      ∑          k      =      0              n      −      1        log  ⁡      (                  3        k        +        1                    3        k        +        3              )    =      ∑          k      =      0              n      −      1        log  ⁡      (    1    −          2              3        k        +        3              )  Now,  log  ⁡      (    1    −          2              3        k        +        3              )    ∼  −      2          3      k      +      3      which is the general term of a divergent series. Hence, by comparison, the series             ∑      log      ⁡              (        1        −                  2                      3            k            +            3                          )             is also divergent, and because it has negative terms, you get that      ∑          k      =      0              +      ∞        log  ⁡      (    1    −          2              3        k        +        3              )    =  −  ∞or equivalently,      lim          n      →      +      ∞        log  ⁡  (      A    n    )  =  −  ∞which leads directly to                                          lim                          n              →              +              ∞                                            A            n                    =          0

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