A (probably easy) question about exchanging the order of this double sum
$begingroup$
I would like to exchange the order of the following:
$sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).
I feel like it should be easy, but so far I am only able to produce
$sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).
In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.
Thanks,
summation
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
$endgroup$
add a comment |
$begingroup$
I would like to exchange the order of the following:
$sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).
I feel like it should be easy, but so far I am only able to produce
$sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).
In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.
Thanks,
summation
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
$endgroup$
add a comment |
$begingroup$
I would like to exchange the order of the following:
$sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).
I feel like it should be easy, but so far I am only able to produce
$sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).
In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.
Thanks,
summation
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
$endgroup$
I would like to exchange the order of the following:
$sum_{k=1}^{i-1}sum_{r=1}^{2k}$ (stuff).
I feel like it should be easy, but so far I am only able to produce
$sum_{r=1}^{2(i-1)}sum_{k=??}^{??}$ (stuff).
In particular, I don't see how to reconcile the inequalities $1 leq rleq 2k,$ and $1 leq k leq i-1$, without introducing half integers into the sum.
Thanks,
summation
summation
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
asked Jan 8 at 6:05
MathIsArtMathIsArt
1328
1328
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
$k$ is always at least $frac r2$ and ranges up to $i-1$
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
$endgroup$
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
|
show 3 more comments
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%2f3065852%2fa-probably-easy-question-about-exchanging-the-order-of-this-double-sum%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
$begingroup$
The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
$k$ is always at least $frac r2$ and ranges up to $i-1$
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
$endgroup$
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
|
show 3 more comments
$begingroup$
The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
$k$ is always at least $frac r2$ and ranges up to $i-1$
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
$endgroup$
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
|
show 3 more comments
$begingroup$
The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
$k$ is always at least $frac r2$ and ranges up to $i-1$
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
$endgroup$
The second sum should be $$sum_{k=lceil frac r2rceil}^{i-1}$$
$k$ is always at least $frac r2$ and ranges up to $i-1$
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
answered Jan 8 at 6:12
Ross MillikanRoss Millikan
300k24200375
300k24200375
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
|
show 3 more comments
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
$begingroup$
I think $k$ starts from $[frac r 2]$ if $r$ is even and $[frac r 2]+1$ if $r$ is odd.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:17
1
1
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
@Kavi That's exactly what $lceil{rover2}rceil$ means.
$endgroup$
– Ivan Neretin
Jan 8 at 6:23
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
I see. I didn't know this. Thanks.
$endgroup$
– Kavi Rama Murthy
Jan 8 at 6:29
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
Your answer is clear, thanks.
$endgroup$
– MathIsArt
Jan 8 at 7:06
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
$begingroup$
@KaviRamaMurthy: those brackets represent the ceiling function-you round up to the next integer. Upside down ones are the floor, where you round down.
$endgroup$
– Ross Millikan
Jan 8 at 15:02
|
show 3 more comments
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%2f3065852%2fa-probably-easy-question-about-exchanging-the-order-of-this-double-sum%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
