If $sum_{n=0}^infty|x_n|^2 $ converges then $sum_{n=0}^infty frac{x_n}{sqrt{n+1}}$ converges?
$begingroup$
Does $(x_n) in ell^2$ imply that $sum_{n=0}^infty frac{x_n}{sqrt{n+1}}$ converges?
I have tried finding an upper bound to the series but have failed so far. I have tried Cauchy-Schwarz, some other futile trials and I tried to find counter examples by hand, but I cannot prove it. I have a strong feeling it is true because it even holds for $x_n=1/n$.
Any hints welcome!
sequences-and-series convergence
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
$endgroup$
add a comment |
$begingroup$
Does $(x_n) in ell^2$ imply that $sum_{n=0}^infty frac{x_n}{sqrt{n+1}}$ converges?
I have tried finding an upper bound to the series but have failed so far. I have tried Cauchy-Schwarz, some other futile trials and I tried to find counter examples by hand, but I cannot prove it. I have a strong feeling it is true because it even holds for $x_n=1/n$.
Any hints welcome!
sequences-and-series convergence
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
$endgroup$
add a comment |
$begingroup$
Does $(x_n) in ell^2$ imply that $sum_{n=0}^infty frac{x_n}{sqrt{n+1}}$ converges?
I have tried finding an upper bound to the series but have failed so far. I have tried Cauchy-Schwarz, some other futile trials and I tried to find counter examples by hand, but I cannot prove it. I have a strong feeling it is true because it even holds for $x_n=1/n$.
Any hints welcome!
sequences-and-series convergence
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
$endgroup$
Does $(x_n) in ell^2$ imply that $sum_{n=0}^infty frac{x_n}{sqrt{n+1}}$ converges?
I have tried finding an upper bound to the series but have failed so far. I have tried Cauchy-Schwarz, some other futile trials and I tried to find counter examples by hand, but I cannot prove it. I have a strong feeling it is true because it even holds for $x_n=1/n$.
Any hints welcome!
sequences-and-series convergence
sequences-and-series convergence
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
asked Jan 21 at 23:16
B.SwanB.Swan
1,0541720
1,0541720
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is not true. Let $x_n=frac 1 {sqrt n ln , n}$ for $n >1$. Then ${x_n} in ell^{2}$ but $sum frac {x_n} {sqrt {n+1} }=infty$. [Use the fact that $sum frac 1 {n (ln, n)^{2}} <infty$ but $sum frac 1 {n (ln, n)}=infty$].
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
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%2f3082542%2fif-sum-n-0-inftyx-n2-converges-then-sum-n-0-infty-fracx-n-sqr%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
$begingroup$
This is not true. Let $x_n=frac 1 {sqrt n ln , n}$ for $n >1$. Then ${x_n} in ell^{2}$ but $sum frac {x_n} {sqrt {n+1} }=infty$. [Use the fact that $sum frac 1 {n (ln, n)^{2}} <infty$ but $sum frac 1 {n (ln, n)}=infty$].
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
add a comment |
$begingroup$
This is not true. Let $x_n=frac 1 {sqrt n ln , n}$ for $n >1$. Then ${x_n} in ell^{2}$ but $sum frac {x_n} {sqrt {n+1} }=infty$. [Use the fact that $sum frac 1 {n (ln, n)^{2}} <infty$ but $sum frac 1 {n (ln, n)}=infty$].
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
add a comment |
$begingroup$
This is not true. Let $x_n=frac 1 {sqrt n ln , n}$ for $n >1$. Then ${x_n} in ell^{2}$ but $sum frac {x_n} {sqrt {n+1} }=infty$. [Use the fact that $sum frac 1 {n (ln, n)^{2}} <infty$ but $sum frac 1 {n (ln, n)}=infty$].
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
$endgroup$
This is not true. Let $x_n=frac 1 {sqrt n ln , n}$ for $n >1$. Then ${x_n} in ell^{2}$ but $sum frac {x_n} {sqrt {n+1} }=infty$. [Use the fact that $sum frac 1 {n (ln, n)^{2}} <infty$ but $sum frac 1 {n (ln, n)}=infty$].
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
answered Jan 21 at 23:24
Kavi Rama MurthyKavi Rama Murthy
62.7k42262
62.7k42262
add a comment |
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%2f3082542%2fif-sum-n-0-inftyx-n2-converges-then-sum-n-0-infty-fracx-n-sqr%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
