Series $sum c_n=s$ , does $nc_nrightarrow0$?
$begingroup$
Series $sum c_n=s$ , does $nc_nrightarrow0$?
I guess it does, but I don't know how to prove , perhaps, for all convergent series , we will have $$c_n=O(frac{1}{nf(n)})$$ here $1/f(n)=o(1)$
calculus
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
$endgroup$
add a comment |
$begingroup$
Series $sum c_n=s$ , does $nc_nrightarrow0$?
I guess it does, but I don't know how to prove , perhaps, for all convergent series , we will have $$c_n=O(frac{1}{nf(n)})$$ here $1/f(n)=o(1)$
calculus
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
$endgroup$
add a comment |
$begingroup$
Series $sum c_n=s$ , does $nc_nrightarrow0$?
I guess it does, but I don't know how to prove , perhaps, for all convergent series , we will have $$c_n=O(frac{1}{nf(n)})$$ here $1/f(n)=o(1)$
calculus
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
$endgroup$
Series $sum c_n=s$ , does $nc_nrightarrow0$?
I guess it does, but I don't know how to prove , perhaps, for all convergent series , we will have $$c_n=O(frac{1}{nf(n)})$$ here $1/f(n)=o(1)$
calculus
calculus
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
asked Dec 15 '18 at 5:15
Alexander LauAlexander Lau
908
908
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No it does not.
Consider
$$
c_n = begin{cases}
dfrac 1n , & sqrt n in mathbb N^*,\
dfrac 1{n^2}, & sqrt n notin mathbb N^*.
end{cases}
$$
Then the series converges, but $n c_n$ is either $1/n$ or $1$, so the limit does not exist.
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
$endgroup$
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
add a comment |
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%2f3040200%2fseries-sum-c-n-s-does-nc-n-rightarrow0%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No it does not.
Consider
$$
c_n = begin{cases}
dfrac 1n , & sqrt n in mathbb N^*,\
dfrac 1{n^2}, & sqrt n notin mathbb N^*.
end{cases}
$$
Then the series converges, but $n c_n$ is either $1/n$ or $1$, so the limit does not exist.
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
$endgroup$
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
add a comment |
$begingroup$
No it does not.
Consider
$$
c_n = begin{cases}
dfrac 1n , & sqrt n in mathbb N^*,\
dfrac 1{n^2}, & sqrt n notin mathbb N^*.
end{cases}
$$
Then the series converges, but $n c_n$ is either $1/n$ or $1$, so the limit does not exist.
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
$endgroup$
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
add a comment |
$begingroup$
No it does not.
Consider
$$
c_n = begin{cases}
dfrac 1n , & sqrt n in mathbb N^*,\
dfrac 1{n^2}, & sqrt n notin mathbb N^*.
end{cases}
$$
Then the series converges, but $n c_n$ is either $1/n$ or $1$, so the limit does not exist.
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
$endgroup$
No it does not.
Consider
$$
c_n = begin{cases}
dfrac 1n , & sqrt n in mathbb N^*,\
dfrac 1{n^2}, & sqrt n notin mathbb N^*.
end{cases}
$$
Then the series converges, but $n c_n$ is either $1/n$ or $1$, so the limit does not exist.
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
answered Dec 15 '18 at 5:20
xbhxbh
6,1901522
6,1901522
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
add a comment |
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
thank you very much !
$endgroup$
– Alexander Lau
Dec 15 '18 at 5:25
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
$c_n=(-1)^{n}frac 1 {sqrt n}$ gives an example where $|nc_n| to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 5:33
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
$begingroup$
Why not post this as an answer?
$endgroup$
– xbh
Dec 15 '18 at 5:34
add a comment |
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%2f3040200%2fseries-sum-c-n-s-does-nc-n-rightarrow0%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
