Particular infinite product convergence
up vote
1
down vote
favorite
My question is a little bit technical but if someone has a clue... I have the following infinite product
$$
P = prod_{n=1}^infty( 1 - q^n(n))
$$
where $q(n)$ is an increasing sequence whose limit for $ntoinfty$ is 1.
If $q(n) = (1/n)^{1/n}$ then $q(n) to 1$ (as $log(n)/nto 0$) and $P to 0$. I am looking for the slowest convergence of the $q(n)$ sequence that makes $P to 0$, or at least examples of convergence slower than the one achieved by $(1/n)^{1/n}to 1$.
Thanks for any help
sequences-and-series infinite-product
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
add a comment |
up vote
1
down vote
favorite
My question is a little bit technical but if someone has a clue... I have the following infinite product
$$
P = prod_{n=1}^infty( 1 - q^n(n))
$$
where $q(n)$ is an increasing sequence whose limit for $ntoinfty$ is 1.
If $q(n) = (1/n)^{1/n}$ then $q(n) to 1$ (as $log(n)/nto 0$) and $P to 0$. I am looking for the slowest convergence of the $q(n)$ sequence that makes $P to 0$, or at least examples of convergence slower than the one achieved by $(1/n)^{1/n}to 1$.
Thanks for any help
sequences-and-series infinite-product
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
My question is a little bit technical but if someone has a clue... I have the following infinite product
$$
P = prod_{n=1}^infty( 1 - q^n(n))
$$
where $q(n)$ is an increasing sequence whose limit for $ntoinfty$ is 1.
If $q(n) = (1/n)^{1/n}$ then $q(n) to 1$ (as $log(n)/nto 0$) and $P to 0$. I am looking for the slowest convergence of the $q(n)$ sequence that makes $P to 0$, or at least examples of convergence slower than the one achieved by $(1/n)^{1/n}to 1$.
Thanks for any help
sequences-and-series infinite-product
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
My question is a little bit technical but if someone has a clue... I have the following infinite product
$$
P = prod_{n=1}^infty( 1 - q^n(n))
$$
where $q(n)$ is an increasing sequence whose limit for $ntoinfty$ is 1.
If $q(n) = (1/n)^{1/n}$ then $q(n) to 1$ (as $log(n)/nto 0$) and $P to 0$. I am looking for the slowest convergence of the $q(n)$ sequence that makes $P to 0$, or at least examples of convergence slower than the one achieved by $(1/n)^{1/n}to 1$.
Thanks for any help
sequences-and-series infinite-product
sequences-and-series infinite-product
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
edited Nov 18 at 19:10
Daniele Tampieri
1,5471619
1,5471619
asked Nov 18 at 18:26
Gianfranco OLDANI
765
asked Nov 18 at 18:26
Gianfranco OLDANI
765
asked Nov 18 at 18:26
Gianfranco OLDANI
765
765
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%2f3003919%2fparticular-infinite-product-convergence%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
