I would like to evaluate: lim n → ∞ 1 n 3 ∑ ℓ = 1 n − 1 ( n 2 − ℓ 2 ) ( n 2 − ( ℓ − 1 ) 2 )

Question

I would like to evaluate:      lim          n      →      ∞            1          n      3            ∑          ℓ      =      1              n      −      1            (          n      2        −          ℓ      2        )    (          n      2        −    (    ℓ    −    1          )      2        )

Zayden Andrade · Accepted Answer

For every   1  ⩽  ℓ  ⩽  n  −  1      n    2    −      ℓ    2    ⩽      (          n      2        −          ℓ      2        )    (          n      2        −    (    ℓ    −    1          )      2        )    ⩽      n    2    −  (  ℓ  −  1      )    2    ,hence the sums       S    n   you are interested in are such that       R    n    ⩽      S    n    ⩽      T    n   for every   n  ⩾  1, with      R    n    =      1          n      3            ∑          ℓ      =      1              n      −      1        (      n    2    −      ℓ    2    )  ,        T    n    =      1          n      3            ∑          ℓ      =      0              n      −      2        (      n    2    −      ℓ    2    )  .The rest should be easy (and the limit is indeed       2    3  )

I would like to evaluate: <munder> <mo movablelimits="true" form="prefix">lim <mrow clas

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New Questions in Calculus 2