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Wissety52

Wissety52

Answered question

2022-05-02

Understanding an Approximation:
1 ( 1 μ n ) μ 1 + ( μ 1 ) 2 μ 3 n .

Answer & Explanation

Tatairfzk

Tatairfzk

Beginner2022-05-03Added 12 answers

Note that μ 1 + ( μ 1 ) 2 = μ ( μ 1 ) .. Using the approximation ( 1 + x ) α 1 + α x for small x, we obtain
( 1 μ n ) μ 1 + ( μ 1 ) 2 1 μ 2 ( μ 1 ) n .
Hence
1 ( 1 μ n ) μ 1 + ( μ 1 ) 2 μ 2 ( μ 1 ) n = μ 3 μ 2 n .
If we can assume that μ is sufficiently large (and indeed we assume that p > 3 log n n so that μ = n p > 3 log n ), then the μ 2 term pales in comparison to μ 3 , so the approximation
μ 3 μ 2 n μ 3 n
can also be made. Another way to view this last approximation is to say that μ 1 μ for large μ so that μ 2 ( μ 1 ) μ 2 ( μ ) = μ 3 ..

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