Nice convergent subsequence of $cos(n)$.
$begingroup$
This question is related to a few questions which have been posted on the website :
- Is there a limit of $cos(n!)$
- Converging subsequence on a circle
- The limit of $sin(n!)$
Because of the irrationality of $pi$, the sequence $(cos(n))_{ninmathbb{N}}$ is dense in $[-1;1]$. For any value $ain[-1;1]$, we can extract a subsequence $(cos(n_k))_{kinmathbb{N}}$ convergent to $a$.
My question is the following:
Does someone know an example of a convergent subsequence of $cos(n)$ with an explicit expression?
Some more comments:
We could define a subsequence in the following way:
$$n_0=1;quad n_{k+1}>n_k text{ such that } |cos(n_k)-a|<1/k.$$
This subsequence is well defined (and unique if we add the condition that $n_{k+1}$ should be minimum) and converges. But I would say, that this not explicit.
I don't have a definition of what should be an explicit expression, and any answer are welcome.
Reading the nice answer of David Speyer in Is there a limit of $cos(n!)$, it seems that we still don't understand enough about $pi$ to proof or disprove that $cos(n!)$ diverges.
Because of these comments, I would not be surprised if the answer to my question is no.
sequences-and-series analysis convergence examples-counterexamples
edited Apr 13 '17 at 12:19
Community♦
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
$endgroup$
add a comment |
$begingroup$
This question is related to a few questions which have been posted on the website :
- Is there a limit of $cos(n!)$
- Converging subsequence on a circle
- The limit of $sin(n!)$
Because of the irrationality of $pi$, the sequence $(cos(n))_{ninmathbb{N}}$ is dense in $[-1;1]$. For any value $ain[-1;1]$, we can extract a subsequence $(cos(n_k))_{kinmathbb{N}}$ convergent to $a$.
My question is the following:
Does someone know an example of a convergent subsequence of $cos(n)$ with an explicit expression?
Some more comments:
We could define a subsequence in the following way:
$$n_0=1;quad n_{k+1}>n_k text{ such that } |cos(n_k)-a|<1/k.$$
This subsequence is well defined (and unique if we add the condition that $n_{k+1}$ should be minimum) and converges. But I would say, that this not explicit.
I don't have a definition of what should be an explicit expression, and any answer are welcome.
Reading the nice answer of David Speyer in Is there a limit of $cos(n!)$, it seems that we still don't understand enough about $pi$ to proof or disprove that $cos(n!)$ diverges.
Because of these comments, I would not be surprised if the answer to my question is no.
sequences-and-series analysis convergence examples-counterexamples
edited Apr 13 '17 at 12:19
Community♦
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
$endgroup$
1
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
1
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01
add a comment |
$begingroup$
This question is related to a few questions which have been posted on the website :
- Is there a limit of $cos(n!)$
- Converging subsequence on a circle
- The limit of $sin(n!)$
Because of the irrationality of $pi$, the sequence $(cos(n))_{ninmathbb{N}}$ is dense in $[-1;1]$. For any value $ain[-1;1]$, we can extract a subsequence $(cos(n_k))_{kinmathbb{N}}$ convergent to $a$.
My question is the following:
Does someone know an example of a convergent subsequence of $cos(n)$ with an explicit expression?
Some more comments:
We could define a subsequence in the following way:
$$n_0=1;quad n_{k+1}>n_k text{ such that } |cos(n_k)-a|<1/k.$$
This subsequence is well defined (and unique if we add the condition that $n_{k+1}$ should be minimum) and converges. But I would say, that this not explicit.
I don't have a definition of what should be an explicit expression, and any answer are welcome.
Reading the nice answer of David Speyer in Is there a limit of $cos(n!)$, it seems that we still don't understand enough about $pi$ to proof or disprove that $cos(n!)$ diverges.
Because of these comments, I would not be surprised if the answer to my question is no.
sequences-and-series analysis convergence examples-counterexamples
edited Apr 13 '17 at 12:19
Community♦
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
$endgroup$
This question is related to a few questions which have been posted on the website :
- Is there a limit of $cos(n!)$
- Converging subsequence on a circle
- The limit of $sin(n!)$
Because of the irrationality of $pi$, the sequence $(cos(n))_{ninmathbb{N}}$ is dense in $[-1;1]$. For any value $ain[-1;1]$, we can extract a subsequence $(cos(n_k))_{kinmathbb{N}}$ convergent to $a$.
My question is the following:
Does someone know an example of a convergent subsequence of $cos(n)$ with an explicit expression?
Some more comments:
We could define a subsequence in the following way:
$$n_0=1;quad n_{k+1}>n_k text{ such that } |cos(n_k)-a|<1/k.$$
This subsequence is well defined (and unique if we add the condition that $n_{k+1}$ should be minimum) and converges. But I would say, that this not explicit.
I don't have a definition of what should be an explicit expression, and any answer are welcome.
Reading the nice answer of David Speyer in Is there a limit of $cos(n!)$, it seems that we still don't understand enough about $pi$ to proof or disprove that $cos(n!)$ diverges.
Because of these comments, I would not be surprised if the answer to my question is no.
sequences-and-series analysis convergence examples-counterexamples
sequences-and-series analysis convergence examples-counterexamples
edited Apr 13 '17 at 12:19
Community♦
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
edited Apr 13 '17 at 12:19
Community♦
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
1
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
asked Feb 17 '14 at 20:45
Gilles BonnetGilles Bonnet
1,180831
1,180831
1
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
1
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01
add a comment |
1
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
1
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01
1
1
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
1
1
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f679984%2fnice-convergent-subsequence-of-cosn%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f679984%2fnice-convergent-subsequence-of-cosn%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
1
$begingroup$
Does "$n_k$ is the numerator of the $k^{rm th}$ convergent in the continued fraction expansion of $2pi$" count as an explicit expression?
$endgroup$
– Micah
Feb 17 '14 at 20:56
1
$begingroup$
@Micah, no. Then I would post a new question: Dos someone knows an explicit expression of "$n_k$, the numerator of the $k^{th}$ convergent in the continued fraction expansion of $2pi$"? ;-) I am looking for something more "explicit".
$endgroup$
– Gilles Bonnet
Feb 17 '14 at 21:01