Find lim n → ∞ n 2 ∑ k = 1 n 1 ( n 2 + k 2 ) ( n 2 + k 2 + n ) .

Question

Find      lim          n      →      ∞            n          2            ∑          k      =      1              n            1          (              n        2            +              k        2            )      (                        n          2                +                  k          2                    +      n      )        .

Jaruckigh · Accepted Answer

For the last integral consider the indefinite integral  I  =  ∫      1                            x          2                +        1            +      1        d  x  =  ∫                              x          2                +        1            −      1              x      2        d  xHere, you can use the hyperbolic trig substitution   x  =  sinh  ⁡  t for the first summand in the numerator which yields  ∫                    x        2            +      1              x      2        d  x  =  ∫      coth    2    ⁡  t     d  t  =  t  −  coth  ⁡  t  +  Cwhich after substituting back yields an antiderivative for I  I  =      sinh          −      1        ⁡  x  −                              x          2                +        1            −      1        x    +  CNote that this function is, as expected, regular at   x  =  0, unlike the one in the 2nd equation above.

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