what the following may equal? sum_(i=1)^(n) [log_2(i)]=?

Mattie Monroe

Mattie Monroe

Answered question

2022-10-13

what the following may equal?
i = 1 n i log 2 ( i ) = ?

Answer & Explanation

faux0101d

faux0101d

Beginner2022-10-14Added 21 answers

Use sum by parts:
1 k n x k Δ y k = x n + 1 y n + 1 x 1 y 1 1 k n y k + 1 Δ x k
In your case Δ y k = log 2 k and x k = k. You'll get a sum similar to the original, you should be able to solve for that one.
caschaillo7

caschaillo7

Beginner2022-10-15Added 1 answers

I could come with a solution with an incisive precision for any base a by doing this:
T ( n ) = i = 1 n i log a ( i ) = [ i = 1 log a ( n ) i ( j = a i 1 + 1 a i j ) ] + log a ( n ) i = a log a ( n ) + 1 n i T ( n ) = [ 1 2 i = 1 log a ( n ) i ( a i 2 ( a 1 ) ( a i + 1 + a i + a ) ) ] + log a ( n ) i = a log a ( n ) + 1 n i Let  T 1 ( n ) = [ 1 2 i = 1 log a ( n ) i ( a i 2 ( a 1 ) ( a i + 1 + a i + a ) ) ] And  T 2 ( n ) = log a ( n ) i = a log a ( n ) + 1 n i T 1 ( n ) = 1 2 ( a 2 1 ) ( ( 1 log a ( n ) ) a 2 log a ( n ) a log a ( n ) + 1 + log a ( n ) a log a ( n ) + 2 + log a ( n ) a 2 log a ( n ) + 2 ( log a ( n ) + 1 ) a log a ( n ) + a + 2 ) T 2 ( n ) = log a ( n ) ( n a log a ( n ) ) ( a log a ( n ) + n + 1 ) 2 Therefore  T ( n ) = T 1 ( n ) + T 2 ( n )

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