Why is $inf g sup g = frac{9}{16} $?
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Consider this question here :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Call that conjecture about $frac{5}{4} $ conjecture $1$.
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16}) ) $
Conjecture $3$ :
——-
Conjecture $2$ is :
$$ sup g(n) space inf g(n) = frac{9}{16} $$
And this follows from conjecture $1$ or vice versa.
——-
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
This question is about the connection ( conjecture $3$).
If you can prove conjecture $1$ or $2$ post it in the other thread.
Btw $ int_0^{2 pi} ln(sin^2(x) + frac{9}{16}) dx = 0 $ indeed as you probably already knew or guessed.
calculus geometry fractions limsup-and-liminf products
asked Nov 15 at 22:45
mick
5,01322063
add a comment |
up vote
2
down vote
favorite
Consider this question here :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Call that conjecture about $frac{5}{4} $ conjecture $1$.
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16}) ) $
Conjecture $3$ :
——-
Conjecture $2$ is :
$$ sup g(n) space inf g(n) = frac{9}{16} $$
And this follows from conjecture $1$ or vice versa.
——-
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
This question is about the connection ( conjecture $3$).
If you can prove conjecture $1$ or $2$ post it in the other thread.
Btw $ int_0^{2 pi} ln(sin^2(x) + frac{9}{16}) dx = 0 $ indeed as you probably already knew or guessed.
calculus geometry fractions limsup-and-liminf products
asked Nov 15 at 22:45
mick
5,01322063
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Consider this question here :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Call that conjecture about $frac{5}{4} $ conjecture $1$.
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16}) ) $
Conjecture $3$ :
——-
Conjecture $2$ is :
$$ sup g(n) space inf g(n) = frac{9}{16} $$
And this follows from conjecture $1$ or vice versa.
——-
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
This question is about the connection ( conjecture $3$).
If you can prove conjecture $1$ or $2$ post it in the other thread.
Btw $ int_0^{2 pi} ln(sin^2(x) + frac{9}{16}) dx = 0 $ indeed as you probably already knew or guessed.
calculus geometry fractions limsup-and-liminf products
asked Nov 15 at 22:45
mick
5,01322063
Consider this question here :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Call that conjecture about $frac{5}{4} $ conjecture $1$.
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16}) ) $
Conjecture $3$ :
——-
Conjecture $2$ is :
$$ sup g(n) space inf g(n) = frac{9}{16} $$
And this follows from conjecture $1$ or vice versa.
——-
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
This question is about the connection ( conjecture $3$).
If you can prove conjecture $1$ or $2$ post it in the other thread.
Btw $ int_0^{2 pi} ln(sin^2(x) + frac{9}{16}) dx = 0 $ indeed as you probably already knew or guessed.
calculus geometry fractions limsup-and-liminf products
calculus geometry fractions limsup-and-liminf products
asked Nov 15 at 22:45
mick
5,01322063
asked Nov 15 at 22:45
mick
5,01322063
asked Nov 15 at 22:45
mick
5,01322063
asked Nov 15 at 22:45
mick
5,01322063
asked Nov 15 at 22:45
mick
5,01322063
5,01322063
add a comment |
add a comment |
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