If $ sumlimits_{n=1}^{N} a_n r_n=0$, what can we say about $ sumlimits_{n=1}^{N} a_n (r_n)^2=0$
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
$endgroup$
add a comment |
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
$endgroup$
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystyle
in titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
add a comment |
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
$endgroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
real-analysis summation
edited Jan 2 at 22:36
ersh
asked Jan 2 at 21:41
ershersh
423113
423113
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystyle
in titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
add a comment |
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystyle
in titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
11
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystyle
in titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystyle
in titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
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%2f3060006%2fif-sum-limits-n-1n-a-n-r-n-0-what-can-we-say-about-sum-limits-n-1%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%2f3060006%2fif-sum-limits-n-1n-a-n-r-n-0-what-can-we-say-about-sum-limits-n-1%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
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystyle
in titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40