logarithmic Series I'm aware that by properties of logarithm sum_(k=1)^n ln (k) = ln (n!)

tramolatzqvg

tramolatzqvg

Answered question

2022-11-17

logarithmic Series
I'm aware that by properties of logarithm
k = 1 n ln ( k ) = ln ( n ! )
My question is if
k = 1 n ln 2 ( k ) = ln 2 ( n ! ) ?
Because when I am verifying the value where n = 5, I get different result... maybe Im missing something... is there a formula that defines
k = 1 n ln 2 ( k )  ?
Here's my computation:
k = 1 n ln 2 ( k ) = ln 2 ( n ! )
k = 1 5 ln 2 ( k ) = ln 2 ( 5 ! )
ln 2 ( 1 ) + ln 2 ( 2 ) + ln 2 ( 3 ) + ln 2 ( 4 ) + ln 2 ( 5 ) = ln 2 ( 120 )
0 + 0.48045 + 1.20694 + 1.92181 + 2.59029 = 22.92007
6.199494 is not equal to 22.92007

Answer & Explanation

Aliya Moore

Aliya Moore

Beginner2022-11-18Added 17 answers

k = 1 n ln 2 k ln 2 n ! because
ln 2 ( x y ) = ( ln x + ln y ) 2 ln 2 x + ln 2 y .

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