Compute the limit lim n → ∞ ∑ k = 1 n ( k n 2 ) k n 2 + 1

Question

Compute the limit      lim          n      →      ∞            ∑          k      =      1              n                  (              k                  n          2                    )                      k                  n          2                    +      1

iceniessyoy · Accepted Answer

Let                    x        =                  k                      n            2                          =        O        (                  1          n                )            and when n is sufficiently large,                    (                  k                      n            2                                    )                                    k                              n                2                                                    =                  e                      x            log            ⁡                          x                                      =        1        +        O        (        x        log        ⁡        x        )            where the big O constant is absolute.                    O        (                  ∑                      k            =            1                                n                                    k                      n            2                                    k                      n            2                          log        ⁡                              k                          n              2                                      )        =        O        (                  ∑                      k            =            1                                n                                    k                      n            2                                    k                      n            2                          log        ⁡                  k                )        +        O        (                  ∑                      k            =            1                                n                                    k                      n            2                                    k                      n            2                          log        ⁡                              n            2                          )        =        o        (        1        )            Hence the principle part of the sum is                              ∑                      k            =            1                                n                                    k                      n            2                          =                  1          2                +        o        (        1        )            Q.E.D.

Compute the limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-T

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