Why is $inf g sup g = frac{9}{16} $?
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
5,03922063
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
5,03922063
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
5,03922063
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
5,03922063
asked Nov 15 at 22:45
mick
5,03922063
asked Nov 15 at 22:45
mick
5,03922063
asked Nov 15 at 22:45
mick
5,03922063
asked Nov 15 at 22:45
mick
5,03922063
5,03922063
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3000441%2fwhy-is-inf-g-sup-g-frac916%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
