Summation of $arccosleft(frac{n^2+r^2+r}{sqrt{(n^2+r^2+r)^2+n^2}}right)$
$begingroup$
I found this question in a book, and cannot solve it.
I have to find the the sum $$S_n=sum_{r=0}^{n-1} arccosleft(frac{n^2+r^2+r}{sqrt{(n^2+r^2+r)^2+n^2}}right)$$
I tried converting this to $arctan(frac{n}{n^2+r^2+r})$ which seemed the most possible way of solving this but can't convert this into a difference of two terms which would help in telescoping the sum.
So my question is:
Am I on the right track or do I need to change my approach completely?
Any help would be highly appreciated.
summation trigonometric-series
edited Dec 17 '18 at 17:46
user10354138
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
$endgroup$
add a comment |
$begingroup$
I found this question in a book, and cannot solve it.
I have to find the the sum $$S_n=sum_{r=0}^{n-1} arccosleft(frac{n^2+r^2+r}{sqrt{(n^2+r^2+r)^2+n^2}}right)$$
I tried converting this to $arctan(frac{n}{n^2+r^2+r})$ which seemed the most possible way of solving this but can't convert this into a difference of two terms which would help in telescoping the sum.
So my question is:
Am I on the right track or do I need to change my approach completely?
Any help would be highly appreciated.
summation trigonometric-series
edited Dec 17 '18 at 17:46
user10354138
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
$endgroup$
add a comment |
$begingroup$
I found this question in a book, and cannot solve it.
I have to find the the sum $$S_n=sum_{r=0}^{n-1} arccosleft(frac{n^2+r^2+r}{sqrt{(n^2+r^2+r)^2+n^2}}right)$$
I tried converting this to $arctan(frac{n}{n^2+r^2+r})$ which seemed the most possible way of solving this but can't convert this into a difference of two terms which would help in telescoping the sum.
So my question is:
Am I on the right track or do I need to change my approach completely?
Any help would be highly appreciated.
summation trigonometric-series
edited Dec 17 '18 at 17:46
user10354138
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
$endgroup$
I found this question in a book, and cannot solve it.
I have to find the the sum $$S_n=sum_{r=0}^{n-1} arccosleft(frac{n^2+r^2+r}{sqrt{(n^2+r^2+r)^2+n^2}}right)$$
I tried converting this to $arctan(frac{n}{n^2+r^2+r})$ which seemed the most possible way of solving this but can't convert this into a difference of two terms which would help in telescoping the sum.
So my question is:
Am I on the right track or do I need to change my approach completely?
Any help would be highly appreciated.
summation trigonometric-series
summation trigonometric-series
edited Dec 17 '18 at 17:46
user10354138
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
edited Dec 17 '18 at 17:46
user10354138
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
edited Dec 17 '18 at 17:46
user10354138
7,4322925
edited Dec 17 '18 at 17:46
user10354138
7,4322925
edited Dec 17 '18 at 17:46
user10354138
7,4322925
7,4322925
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
asked Dec 17 '18 at 17:19
user35508user35508
2,9392623
2,9392623
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: compute $$tanleft(arctan{frac{r+1}{n}}-arctan{frac{r}{n}}right).$$
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
$endgroup$
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
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%2f3044204%2fsummation-of-arccos-left-fracn2r2r-sqrtn2r2r2n2-right%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
$begingroup$
Hint: compute $$tanleft(arctan{frac{r+1}{n}}-arctan{frac{r}{n}}right).$$
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
$endgroup$
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
add a comment |
$begingroup$
Hint: compute $$tanleft(arctan{frac{r+1}{n}}-arctan{frac{r}{n}}right).$$
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
$endgroup$
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
add a comment |
$begingroup$
Hint: compute $$tanleft(arctan{frac{r+1}{n}}-arctan{frac{r}{n}}right).$$
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
$endgroup$
Hint: compute $$tanleft(arctan{frac{r+1}{n}}-arctan{frac{r}{n}}right).$$
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
answered Dec 17 '18 at 17:32
MindlackMindlack
4,920211
4,920211
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
add a comment |
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
$begingroup$
Ah ,That was it...Thanks,I got it
$endgroup$
– user35508
Dec 17 '18 at 17:35
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%2f3044204%2fsummation-of-arccos-left-fracn2r2r-sqrtn2r2r2n2-right%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
