How to compute <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAt

Rachel Villa

Rachel Villa

Answered question

2022-05-19

How to compute lim n 1 n [ ( n 2 + 1 2 ) ( n 2 + 2 2 ) 2 ( n 2 + n 2 ) n ] 1 n 2

Answer & Explanation

Tristan Ward

Tristan Ward

Beginner2022-05-20Added 8 answers

We have
1 n ( k = 1 n ( n 2 + k 2 ) k ) 1 / n 2 = 1 n exp ( 1 n 2 k = 1 n k log ( n 2 + k 2 ) ) = 1 n exp ( 1 n k = 1 n k n log ( 1 + ( k n ) 2 ) + 2 log n n 2 k = 1 n k ) = exp ( 1 n k = 1 n k n log ( 1 + ( k n ) 2 ) + n ( n + 1 ) n 2 log n log n ) = exp ( 1 n k = 1 n k n log ( 1 + ( k n ) 2 ) + log n n ) exp ( 0 1 x log ( 1 + x 2 ) d x + 0 ) = exp ( log 2 1 2 ) = 2 e 1 / 2 .

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