I was wondering whether this limit converges to zero: lim n → ∞ ∑ i = 0 n i n 2 = 0

Question

I was wondering whether this limit converges to zero:      lim          n      →      ∞                          ∑                  i          =          0                          n                                      i                            n      2        =  0

gudstrufy47j · Accepted Answer

The sequence in the question can be written as      b    n    =            a      n              n              where         a    n    =      (                  1        n              +                  2        n              +    ⋯    +                  n        n              )        1    n    But       a    n   can be taken as a Riemann sum and      lim          n      →      ∞            a    n    =      ∫    0    1        x    d  xIt follows that      lim          n      →      ∞                  a      n              n        =      lim          n      →      ∞            a    n        lim          n      →      ∞            1          n        =  0

Jay Barrett · Accepted Answer

You can make very good approximations using generalized harmonic numbers      ∑          i      =      0        n        i    =      H    n                  (        −                  1          2                )            For large values of n, then      ∑          i      =      0        n        i    =            2              n                  3                      /                    2                      3    +            n        2    +  ζ      (    −          1      2        )    +  O      (          1              n                  3                      /                    2                      )        1          n      2            ∑          i      =      0        n        i    =      2          3              n              +      1          2      n              n              +  ⋯Use it for   n  =  100; the trucated series gives       403    6000    =  0.0671667 while the exact value is   0.0671463 (relative error of   0.03%)

I was wondering whether this limit converges to zero: <munder> <mo movablelimits="true" form

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New Questions in Calculus 1