Kendrick Hampton

Kendrick Hampton

Answered

2022-06-25

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

Do you have a similar question?

Recalculate according to your conditions!

Answer & Explanation

laure6237ma

laure6237ma

Expert

2022-06-26Added 27 answers

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 ) .

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