Compute lim n → ∞ ( 3 n ) 1 / 3 x n

Question

Compute       lim          n      →      ∞        (  3  n      )          1              /            3            x    n

yelashwag8 · Accepted Answer

We have that      x    n    →  0  ,          S    n    =      ∑          k      =      1              n            x    k    2    →  ∞We use Stolz&#039;s lemma to show      lim    n              3      n              S      n      3        =  1And we would be done after that, indeed      lim    n    3  n      x    n    3    =      lim    n              3      n              x        n        3                            x        n        3                    S        n        3              =      lim    n              3      n              S      n      3        =  1and so by continuity of   f  (  x  )  =      x          1      3       we are done.Now let&#039;s prove our claim      lim    n              3      n              S      n      3        =      lim    n              3      (      n      +      1      )      −      3      n                      S                  n          +          1                3            −              S        n        3              =  =      3          (              S                  n          +          1                    −              S                  n                    )      (              S                  n          +          1                2            +              S                  n          +          1                            S        n            +              S        n        2            )        =  =      3                  x                  n          +          1                2            (              S                  n          +          1                2            +              S                  n          +          1                            S        n            +              S        n        2            )        =  =      3                  x                  n          +          1                2                    S                  n          +          1                2            (      1      +                        S          n                          S                      n            +            1                              +                        S          n                          S                      n            +            1                                      )        2              =Now      lim    n              S      n              S              n        +        1              =      lim          n            (    1    −                  x                  n          +          1                2                    S                  n          +          1                      )    =  1Because            x      n      2              S      n        =            x      n      3                      S        n                    x        n            So we are left      lim    n              3      n              S      n      3        =      lim    n        3                  x                  n          +          1                2                    S                  n          +          1                2                  1    3    =  1

Compute lim_(n=> infty)(3n)^(1/3)x_n

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Answer & Explanation

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