lim n → ∞ n 2 ( ln ⁡ ( n + 1 ) − ln ⁡ n ) = 0But I haven't any ideas how to do it... My calculations shows that this sequence is monotonously decreasing.I've proved that using inequality ln ⁡ ( 1 + x a ) ≤ a x and double-sided theorem.

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lim          n      →      ∞        n      2                (            ln    ⁡    (    n    +    1    )    −      ln    ⁡    n              )        =  0But I haven&#039;t any ideas how to do it... My calculations shows that this sequence is monotonously decreasing.I&#039;ve proved that using inequality   ln  ⁡  (  1  +      x    a    )  ≤  a  x and double-sided theorem.

furniranizq · Accepted Answer

n      2        (          ln      ⁡      (      n      +      1      )        −          ln      ⁡      n        )    =      2    n            ln      ⁡      (      n      +      1      )      −      ln      ⁡      n                      ln        ⁡        (        n        +        1        )            +              ln        ⁡        n                =      2    n            ln      ⁡              (        1        +                  1          n                )                            ln        ⁡        (        n        +        1        )            +              ln        ⁡        n                  ∼    ∞        2    n            1      n              2              ln        ⁡        n              =      1          2      ln      ⁡      n        →  0

letumsnemesislh · Accepted Answer

Since      a    n    :=  n      2        (          ln      ⁡      (      n      +      1      )        −          ln      ⁡      n        )    =      2    n            ln      ⁡      (      n      +      1      )      −      ln      ⁡      n                      ln        ⁡        (        n        +        1        )            +              ln        ⁡        n              =      2    n            ln      ⁡              (        1        +                  1          n                )                            ln        ⁡        (        n        +        1        )            +              ln        ⁡        n              ,we have      a    n    =      2              n              (                  1          n                −                  1                      n            2                          +                  1                      n            3                          −        …        )                            ln        ⁡        (        n        +        1        )            +              ln        ⁡        n              =      2              1      −              1        n            +              1                  n          2                    −      …                      ln        ⁡        (        n        +        1        )            +              ln        ⁡        n            it follows that      lim          n      →      ∞            a    n    =  0.

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