Closed form for a series involving the $Gamma$ and $zeta$ functions
$begingroup$
I was just wondering wether one can derive a closed form for $$sum_{n=1}^{infty}frac{1}{Gammaleft(frac{1}{n}right)zetaleft(1+frac{1}{n}right)}$$
Numerical simulation gives $S=1.20154...$
The product formula between $Gamma$ and $zeta$ gives $$sum_{n=1}^{infty}frac{1}{ncdot a_n} text{ with } a_n=int_{0}^{infty}frac{x^{frac{1}{n}}}{e^x-1}text{d}x$$
But I can't seem to go any further.
Any suggestion ? Is it even doable ?
real-analysis calculus sequences-and-series analysis summation
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
$endgroup$
add a comment |
$begingroup$
I was just wondering wether one can derive a closed form for $$sum_{n=1}^{infty}frac{1}{Gammaleft(frac{1}{n}right)zetaleft(1+frac{1}{n}right)}$$
Numerical simulation gives $S=1.20154...$
The product formula between $Gamma$ and $zeta$ gives $$sum_{n=1}^{infty}frac{1}{ncdot a_n} text{ with } a_n=int_{0}^{infty}frac{x^{frac{1}{n}}}{e^x-1}text{d}x$$
But I can't seem to go any further.
Any suggestion ? Is it even doable ?
real-analysis calculus sequences-and-series analysis summation
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
$endgroup$
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52
add a comment |
$begingroup$
I was just wondering wether one can derive a closed form for $$sum_{n=1}^{infty}frac{1}{Gammaleft(frac{1}{n}right)zetaleft(1+frac{1}{n}right)}$$
Numerical simulation gives $S=1.20154...$
The product formula between $Gamma$ and $zeta$ gives $$sum_{n=1}^{infty}frac{1}{ncdot a_n} text{ with } a_n=int_{0}^{infty}frac{x^{frac{1}{n}}}{e^x-1}text{d}x$$
But I can't seem to go any further.
Any suggestion ? Is it even doable ?
real-analysis calculus sequences-and-series analysis summation
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
$endgroup$
I was just wondering wether one can derive a closed form for $$sum_{n=1}^{infty}frac{1}{Gammaleft(frac{1}{n}right)zetaleft(1+frac{1}{n}right)}$$
Numerical simulation gives $S=1.20154...$
The product formula between $Gamma$ and $zeta$ gives $$sum_{n=1}^{infty}frac{1}{ncdot a_n} text{ with } a_n=int_{0}^{infty}frac{x^{frac{1}{n}}}{e^x-1}text{d}x$$
But I can't seem to go any further.
Any suggestion ? Is it even doable ?
real-analysis calculus sequences-and-series analysis summation
real-analysis calculus sequences-and-series analysis summation
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
edited Dec 13 '18 at 14:07
Harmonic Sun
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
asked Dec 13 '18 at 14:01
Harmonic SunHarmonic Sun
6529
6529
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52
add a comment |
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52
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%2f3038031%2fclosed-form-for-a-series-involving-the-gamma-and-zeta-functions%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%2f3038031%2fclosed-form-for-a-series-involving-the-gamma-and-zeta-functions%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
$begingroup$
$S=1.20652522381311998892341483649465039450107654831603ldots$
$endgroup$
– metamorphy
Dec 29 '18 at 9:52