I would like to find an equivalent of u n − u ∞ = ∑ k = 1 n n n 2 + k 2 − u ∞

Question

I would like to find an equivalent of      u          n        −      u          ∞        =      ∑          k      =      1              n            n                  n        2            +              k        2              −      u          ∞

laure6237ma · Accepted Answer

Your numerical work indeed leads to the right conjecture       u          n        −      π    4    ∼  −      1          4      n      . I am feeling lazy, so to prove the result I will appeal to a standard result about   TRAP  (  n  ), the Trapezoidal Rule with division into n equal parts. It is known that under suitable differentiability assumptions, which are amply met here, the error in   TRAP  (  n  ) is   O  (  1      /        n    2    ). Note that  TRAP  (  n  )  =      ∑          k      =      1              n      −      1            n                  n        2            +              k        2              +      1    2        (          n              n        2              +          n              2                  n          2                      )    .Thus  TRAP  (  n  )  =      ∑          k      =      1              n            n                  n        2            +              k        2              +      1    2        (          n              n        2              +          n              2                  n          2                      )    −      n          2              n        2              =      ∑          k      =      1              n            n                  n        2            +              k        2              +      1          4      n        .It follows that      ∑          k      =      1              n            n                  n        2            +              k        2              =      π    4    −      1          4      n        +  O  (  1      /        n    2    )  .

I would like to find an equivalent of u <mrow class="MJX-TeXAtom-ORD"> n </mr

Answered question

Answer & Explanation

New Questions in Calculus 2