Question about convergence of infinite productsInfinite products (involving complex numbers)Convergence of a productHelp with Convergence/DivergenceA basic sequence convergence questionQuestion about series increasing and convergenceProof about the convergence of a sequenceStatements regarding sequences and seriesInfinite series: Integral test( Proof )Question regarding implication of series convergence.Convergence of the ratio of two infinite series.
Multi tool use
C++ check if statement can be evaluated constexpr
Are Captain Marvel's powers affected by Thanos breaking the Tesseract and claiming the stone?
Non-trope happy ending?
How can ping know if my host is down
What is going on with gets(stdin) on the site coderbyte?
Does the Linux kernel need a file system to run?
Does Doodling or Improvising on the Piano Have Any Benefits?
Make a Bowl of Alphabet Soup
Giving feedback to someone without sounding prejudiced
Why do Radio Buttons not fill the entire outer circle?
How to make money from a browser who sees 5 seconds into the future of any web page?
Is there a nicer/politer/more positive alternative for "negates"?
Can I say "fingers" when referring to toes?
I found an audio circuit and I built it just fine, but I find it a bit too quiet. How do I amplify the output so that it is a bit louder?
Why can't the Brexit deadlock in the UK parliament be solved with a plurality vote?
Multiplicative persistence
Is there any evidence that Cleopatra and Caesarion considered fleeing to India to escape the Romans?
Why do ¬, ∀ and ∃ have the same precedence?
Change the color of a single dot in `ddot` symbol
Can I turn my anal-retentiveness into a career?
How would you translate "more" for use as an interface button?
How to convince somebody that he is fit for something else, but not this job?
Quoting Keynes in a lecture
What is Cash Advance APR?
Question about convergence of infinite products
Infinite products (involving complex numbers)Convergence of a productHelp with Convergence/DivergenceA basic sequence convergence questionQuestion about series increasing and convergenceProof about the convergence of a sequenceStatements regarding sequences and seriesInfinite series: Integral test( Proof )Question regarding implication of series convergence.Convergence of the ratio of two infinite series.
$begingroup$
Because $lim_nrightarrow infty(1+fracxn)^n = e^x$, is it true that $prod_n(1+x_n)$ is bounded above by $e^sum_n=1^infty x_n$ and hence $prod_n(1+x_n)$ converges or diverges depending on $sum_n=1^inftyx_n$? Thanks!
calculus sequences-and-series
asked Mar 14 at 16:18
manifoldedmanifolded
48819
$endgroup$
add a comment |
$begingroup$
Because $lim_nrightarrow infty(1+fracxn)^n = e^x$, is it true that $prod_n(1+x_n)$ is bounded above by $e^sum_n=1^infty x_n$ and hence $prod_n(1+x_n)$ converges or diverges depending on $sum_n=1^inftyx_n$? Thanks!
calculus sequences-and-series
asked Mar 14 at 16:18
manifoldedmanifolded
48819
$endgroup$
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11
add a comment |
$begingroup$
Because $lim_nrightarrow infty(1+fracxn)^n = e^x$, is it true that $prod_n(1+x_n)$ is bounded above by $e^sum_n=1^infty x_n$ and hence $prod_n(1+x_n)$ converges or diverges depending on $sum_n=1^inftyx_n$? Thanks!
calculus sequences-and-series
asked Mar 14 at 16:18
manifoldedmanifolded
48819
$endgroup$
Because $lim_nrightarrow infty(1+fracxn)^n = e^x$, is it true that $prod_n(1+x_n)$ is bounded above by $e^sum_n=1^infty x_n$ and hence $prod_n(1+x_n)$ converges or diverges depending on $sum_n=1^inftyx_n$? Thanks!
calculus sequences-and-series
calculus sequences-and-series
asked Mar 14 at 16:18
manifoldedmanifolded
48819
asked Mar 14 at 16:18
manifoldedmanifolded
48819
edited Mar 14 at 17:08
manifolded
asked Mar 14 at 16:18
manifoldedmanifolded
48819
asked Mar 14 at 16:18
manifoldedmanifolded
48819
asked Mar 14 at 16:18
manifoldedmanifolded
48819
48819
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11
add a comment |
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $|x_n|<1$, $$sum_n=1^mlog(1+x_n)=sum_n=1^mx_n-frac12sum_n=1^mx_n^2+frac13sum_n=1^mx_n^3-cdots$$
and there is no reason that
$$lim_mtoinftysum_n=1^mlog(1+x_n)=lim_mtoinftysum_n=1^mx_n.$$
Given the change of the question, this answer is irrelevant.
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
$endgroup$
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%2f3148206%2fquestion-about-convergence-of-infinite-products%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
$begingroup$
If $|x_n|<1$, $$sum_n=1^mlog(1+x_n)=sum_n=1^mx_n-frac12sum_n=1^mx_n^2+frac13sum_n=1^mx_n^3-cdots$$
and there is no reason that
$$lim_mtoinftysum_n=1^mlog(1+x_n)=lim_mtoinftysum_n=1^mx_n.$$
Given the change of the question, this answer is irrelevant.
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
If $|x_n|<1$, $$sum_n=1^mlog(1+x_n)=sum_n=1^mx_n-frac12sum_n=1^mx_n^2+frac13sum_n=1^mx_n^3-cdots$$
and there is no reason that
$$lim_mtoinftysum_n=1^mlog(1+x_n)=lim_mtoinftysum_n=1^mx_n.$$
Given the change of the question, this answer is irrelevant.
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
If $|x_n|<1$, $$sum_n=1^mlog(1+x_n)=sum_n=1^mx_n-frac12sum_n=1^mx_n^2+frac13sum_n=1^mx_n^3-cdots$$
and there is no reason that
$$lim_mtoinftysum_n=1^mlog(1+x_n)=lim_mtoinftysum_n=1^mx_n.$$
Given the change of the question, this answer is irrelevant.
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
$endgroup$
If $|x_n|<1$, $$sum_n=1^mlog(1+x_n)=sum_n=1^mx_n-frac12sum_n=1^mx_n^2+frac13sum_n=1^mx_n^3-cdots$$
and there is no reason that
$$lim_mtoinftysum_n=1^mlog(1+x_n)=lim_mtoinftysum_n=1^mx_n.$$
Given the change of the question, this answer is irrelevant.
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
answered Mar 14 at 17:09
Yves DaoustYves Daoust
131k676229
131k676229
add a comment |
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%2f3148206%2fquestion-about-convergence-of-infinite-products%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
GHSiIN,juEO4dEw u86b,7bhf0 B,V3bS,eAf 1s 4,UpB
$begingroup$
What do you mean by $$lim_ntoinftyprod_nleft(1+x_nright)$$ exactly ?
$endgroup$
– Yves Daoust
Mar 14 at 16:52
$begingroup$
$prod_n (1+x_n) = (1+x_1)(1+x_2)..$ and by the limit I mean, is this product bounded above by $e^sum x_n$
$endgroup$
– manifolded
Mar 14 at 17:06
$begingroup$
"bounded above" makes it a very different question !
$endgroup$
– Yves Daoust
Mar 14 at 17:10
$begingroup$
True, my bad, don't know why I did that. Thanks for the answer.
$endgroup$
– manifolded
Mar 14 at 17:11