Prove that for $vert xvert le 1$ $sum_{n=1}^{infty} frac {sin n}{n} x^n=frac 12 i left(ln (1-xe^i)-...
$begingroup$
Prove that for $vert xvert lt 1$ $$sum_{n=1}^{infty} frac {sin n}{n} x^n=frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
My try:
I solved this using the fact that $$sin x=frac {e^{ix}-e^{-ix}}{2i}$$
Hence $$sum_{n=1}^{infty} frac {sin n}{n} x^n=sum_{n=1}^{infty} frac {e^{in} -e^{-in}}{2in}=frac {-i}{2}left(sum_{n=1}^{infty} frac {(xe^i)^n. }{n}-sum_{n=1}^{infty} frac {(xe^{-i})^n}{n}right) =frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
I wanted to know if there could be any other alternative approach.
calculus sequences-and-series trigonometry
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
$endgroup$
add a comment |
$begingroup$
Prove that for $vert xvert lt 1$ $$sum_{n=1}^{infty} frac {sin n}{n} x^n=frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
My try:
I solved this using the fact that $$sin x=frac {e^{ix}-e^{-ix}}{2i}$$
Hence $$sum_{n=1}^{infty} frac {sin n}{n} x^n=sum_{n=1}^{infty} frac {e^{in} -e^{-in}}{2in}=frac {-i}{2}left(sum_{n=1}^{infty} frac {(xe^i)^n. }{n}-sum_{n=1}^{infty} frac {(xe^{-i})^n}{n}right) =frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
I wanted to know if there could be any other alternative approach.
calculus sequences-and-series trigonometry
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
$endgroup$
add a comment |
$begingroup$
Prove that for $vert xvert lt 1$ $$sum_{n=1}^{infty} frac {sin n}{n} x^n=frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
My try:
I solved this using the fact that $$sin x=frac {e^{ix}-e^{-ix}}{2i}$$
Hence $$sum_{n=1}^{infty} frac {sin n}{n} x^n=sum_{n=1}^{infty} frac {e^{in} -e^{-in}}{2in}=frac {-i}{2}left(sum_{n=1}^{infty} frac {(xe^i)^n. }{n}-sum_{n=1}^{infty} frac {(xe^{-i})^n}{n}right) =frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
I wanted to know if there could be any other alternative approach.
calculus sequences-and-series trigonometry
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
$endgroup$
Prove that for $vert xvert lt 1$ $$sum_{n=1}^{infty} frac {sin n}{n} x^n=frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
My try:
I solved this using the fact that $$sin x=frac {e^{ix}-e^{-ix}}{2i}$$
Hence $$sum_{n=1}^{infty} frac {sin n}{n} x^n=sum_{n=1}^{infty} frac {e^{in} -e^{-in}}{2in}=frac {-i}{2}left(sum_{n=1}^{infty} frac {(xe^i)^n. }{n}-sum_{n=1}^{infty} frac {(xe^{-i})^n}{n}right) =frac 12 i left(ln (1-xe^i)- ln(1-xe^{-i})right) $$
I wanted to know if there could be any other alternative approach.
calculus sequences-and-series trigonometry
calculus sequences-and-series trigonometry
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
edited Dec 8 '18 at 11:37
Digamma
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
asked Dec 8 '18 at 11:32
DigammaDigamma
6,1671440
6,1671440
add a comment |
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%2f3030989%2fprove-that-for-vert-x-vert-le-1-sum-n-1-infty-frac-sin-nn-xn%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%2f3030989%2fprove-that-for-vert-x-vert-le-1-sum-n-1-infty-frac-sin-nn-xn%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
