How to prove the convergence of the following series The Next CEO of Stack OverflowConvergence of alternating series not subject to alternating series testSeries and uniform convergenceProve series convergenceBig O and little o, absolute convergence of series where $a_n = O(b_n) $ or $a_n = o(b_n)$Convergence of a sum of seriesProve if $sumlimits_n=1^ infty a_n$ converges, $b_n$ is bounded & monotone, then $sumlimits_n=1^ infty a_nb_n$ converges.A question about real series $sum_n=1^infty a_n$ and $sum_n=1^infty b_n$Proof of convergence of sequences and Fourier series convergenceConvergence of series $sum_n=1^infty frac1+x^2nn^6$Help with convergence tests for series
What flight has the highest ratio of time difference to flight time?
What can we do to stop prior company from asking us questions?
Complex fractions
Can I equip Skullclamp on a creature I am sacrificing?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
Spaceship fuel on Europa
How do we know the LHC results are robust?
Should I tutor a student who I know has cheated on their homework?
Unreliable Magic - Is it worth it?
Why is there a PLL in CPU?
Why didn't Khan get resurrected in the Genesis Explosion?
Disadvantage of gaining multiple levels at once in a short milestone-XP game
Is HostGator storing my password in plaintext?
Different harmonic changes implied by a simple descending scale
Inappropriate reference requests from Journal reviewers
What is the purpose of the Evocation wizard's Potent Cantrip feature?
Beyond letters and diaries—exercises to explore characters' personalities and motivation
What was the first Unix version to run on a microcomputer?
Is it ever safe to open a suspicious html file (e.g. email attachment)?
Would a galaxy be visible from outside, but nearby?
What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
Contours of a clandestine nature
FBX seems to be empty when imported into Blender
Was a professor correct to chastise me for writing "Prof. X" rather than "Professor X"?
How to prove the convergence of the following series
The Next CEO of Stack OverflowConvergence of alternating series not subject to alternating series testSeries and uniform convergenceProve series convergenceBig O and little o, absolute convergence of series where $a_n = O(b_n) $ or $a_n = o(b_n)$Convergence of a sum of seriesProve if $sumlimits_n=1^ infty a_n$ converges, $b_n$ is bounded & monotone, then $sumlimits_n=1^ infty a_nb_n$ converges.A question about real series $sum_n=1^infty a_n$ and $sum_n=1^infty b_n$Proof of convergence of sequences and Fourier series convergenceConvergence of series $sum_n=1^infty frac1+x^2nn^6$Help with convergence tests for series
$begingroup$
Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.
If $a_n$ is positive, I have been able to solve. How we can solve in general?
real-analysis sequences-and-series
asked 1 hour ago
J.DoeJ.Doe
542
$endgroup$
add a comment |
$begingroup$
Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.
If $a_n$ is positive, I have been able to solve. How we can solve in general?
real-analysis sequences-and-series
asked 1 hour ago
J.DoeJ.Doe
542
$endgroup$
add a comment |
$begingroup$
Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.
If $a_n$ is positive, I have been able to solve. How we can solve in general?
real-analysis sequences-and-series
asked 1 hour ago
J.DoeJ.Doe
542
$endgroup$
Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.
If $a_n$ is positive, I have been able to solve. How we can solve in general?
real-analysis sequences-and-series
real-analysis sequences-and-series
asked 1 hour ago
J.DoeJ.Doe
542
asked 1 hour ago
J.DoeJ.Doe
542
asked 1 hour ago
J.DoeJ.Doe
542
asked 1 hour ago
J.DoeJ.Doe
542
asked 1 hour ago
J.DoeJ.Doe
542
542
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.
It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.
But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
$endgroup$
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%2f3167363%2fhow-to-prove-the-convergence-of-the-following-series%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
$begingroup$
Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.
It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.
But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
$endgroup$
add a comment |
$begingroup$
Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.
It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.
But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
$endgroup$
add a comment |
$begingroup$
Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.
It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.
But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
$endgroup$
Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.
It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.
But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
answered 1 hour ago
Eclipse SunEclipse Sun
7,9451438
7,9451438
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%2f3167363%2fhow-to-prove-the-convergence-of-the-following-series%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
