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Boilanubjaini8f

Boilanubjaini8f

Answered question

2022-06-05

i = 1 k 1 ( 1 i 2 N ) 1 ( k 2 ) 2 N
what assumption are needed for this approximation to be useful.

Answer & Explanation

Abigail Palmer

Abigail Palmer

Beginner2022-06-06Added 30 answers

Let P be the product; then
log P = i = 1 k 1 log ( 1 i 2 N )
When N is sufficiently large, then we may use the approximation log ( 1 y ) y and get
log ( 1 y ) y
log P 1 2 N i = 1 k 1 i = 1 2 N k ( k 1 ) 2 = 1 2 N ( k 2 )
Thus,
P e 1 2 N ( k 2 ) 1 1 2 N ( k 2 )
The above holds when
1 2 1 4 N 2 ( k 2 ) 2
is sufficiently small.
For the 1st "sufficient", we need
1 2 1 4 N 2 i = 1 k 1 i 2 = 1 8 N 2 ( k 1 ) k ( 2 k 1 ) 6 k 3 24 N 2
or k < C 1 N 2 / 3 , where C 1 is a small number. For the second sufficient, we need k 4 / ( 32 N 2 ) to be small, or k < C 2 N . For the estimate to hold overall, we need the smaller of the two estimates, or that k < C 2 N .

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