I need to solve the following problem: lim n ↦ ∞ ( n n − 1 ) 1 n .

Question

I need to solve the following problem:      lim          n      ↦      ∞        (      n    n    −  1      )                  1        n              .

Kaydence Washington · Accepted Answer

Let n be a positive integer.Using Landau notations,       n    n    −  1  =  exp  ⁡  (      1    n    ln  ⁡  (  n  )  )  −  1  =            ln      ⁡      (      n      )        n    +  o      (                  ln        ⁡        (        n        )            n        )   as   n  →  +  ∞Then       1    n    ln  ⁡  (      n    n    −  1  )  =      1    n    ln  ⁡      (                  ln        ⁡        (        n        )            n        +    o          (                        ln          ⁡          (          n          )                n            )        )    →  0 when   n  →  +  ∞And   (      n    n    −  1      )          1      n        =  exp  ⁡  (      1    n    ln  ⁡  (      n    n    −  1  )  )  →  exp  ⁡  (  0  )  =  1 when   n  →  +  ∞

Celia Lucas · Accepted Answer

We have for n&amp;gt;2            ln      ⁡      (              n        n            −      1      )        n    =            ln      ⁡              [                  n          n                          (          1          −                      1                          n              n                                )                ]              n    =            ln      ⁡      (              n        n            )        n    +            ln      ⁡              (        1        −                  1                      n            n                          )              n    =            ln      ⁡      (      n      )              n      2        +            ln      ⁡              (        1        −                  1                      n            n                          )              n  But for x&amp;gt;1,   ln  ⁡  (  x  )  ≤  x  −  1 so  0  ≤            ln      ⁡      (      n      )              n      2        ≤            n      −      1              n      2      and the sequence             (      ln      ⁡              (        1        −                  1                      n            n                          )            )              n      ≥      2       is bounded

I need to solve the following problem: <munder> <mo movablelimits="true" form="prefix">lim

Answered question

Answer & Explanation

New Questions in Calculus 1