What is the asymptotic behavior of the sequence: s n = ∑ k = 1 n k 1 / 4

Question

What is the asymptotic behavior of the sequence:      s    n    =      ∑          k      =      1              n            k          1              /            4

Mihevcekd · Accepted Answer

Let   k  ⩾  1. Since the function   x  ↦      x          1              /            4       is increasing,   (  k  −  1      )          1              /            4        ⩽      x          1              /            4        ⩽      k          1              /            4       for every   k  −  1  ⩽  x  ⩽  k. Integrating this double inequality yields  (  k  −  1      )          1              /            4        ⩽      ∫          k      −      1        k        x          1              /            4            d    x  ⩽      k          1              /            4        .Summing these from   k  =  1 to   k  =  n yields      s          n      −      1        =      ∑          k      =      0              n      −      1            k          1              /            4        ⩽      ∫          0        n        x          1              /            4            d    x  ⩽      ∑          k      =      1              n            k          1              /            4        =      s    n    .Since the integral is             4      5            n          5              /            4       and       s          n      −      1        =      s    n    −      n          1              /            4      , this yields, for every   n  ⩾  1            4      5            n          5              /            4        ⩽      s    n    ⩽            4      5            n          5              /            4        +      n          1              /            4        .Note finally that       n          1              /            4        =  o  (      n          5              /            4        ) hence a (much weakened) version of this is       s    n    ∼            4      5            n          5              /            4

What is the asymptotic behavior of the sequence: s n </msub> = <munderover> &

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