How would you go about solving this limit knowing it's result but no way of proving it. lim n → ∞ ( n − ∑ k = 1 n ( e k n 2 ) )

Question

How would you go about solving this limit knowing it&#039;s result but no way of proving it.      lim          n      →      ∞            (    n    −          ∑              k        =        1            n              (              e                              k                          n              2                                          )            )

g2joey15 · Accepted Answer

By Taylor&#039;s theorem,      e                  k                  n          2                      =  1  +      k          n      2        +  O      (                  (                  k                      n            2                          )            2        )    =  1  +      k          n      2        +  O      (          1              n        2              )    .Then                              lim                      n            →            ∞                                    ∑                      k            =            1                    n                          (          1          −                      e                                          k                                  n                  2                                                              )                                    =                  lim                      n            →            ∞                                    ∑                      k            =            1                    n                          (          −                      k                          n              2                                +          O                      (                          1                              n                2                                      )                    )                                                  =        −                  lim                      n            →            ∞                                    ∑                      k            =            1                    n                          k                      n            2                                                            =        −                  lim                      n            →            ∞                                    1          n                          ∑                      k            =            1                    n                          k          n                                                  =        −                  ∫          0          1                x                d        x                                          =        −                  1          2                .

How would you go about solving this limit knowing it's result but no way of proving it. <munder>

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