Counterexamples related to a convergent positive series












0












$begingroup$



Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:38










  • $begingroup$
    You should add these to your post.
    $endgroup$
    – xbh
    Dec 11 '18 at 6:44










  • $begingroup$
    $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:47










  • $begingroup$
    @mathworker21 no, we have to come up with counter-examples for these statements.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:50










  • $begingroup$
    @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:51


















0












$begingroup$



Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:38










  • $begingroup$
    You should add these to your post.
    $endgroup$
    – xbh
    Dec 11 '18 at 6:44










  • $begingroup$
    $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:47










  • $begingroup$
    @mathworker21 no, we have to come up with counter-examples for these statements.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:50










  • $begingroup$
    @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:51
















0












0








0





$begingroup$



Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series







share|cite|improve this question











$endgroup$





Let ${a_n}$ be a sequence such that $a_n > 0$ for all $n ≥ 1$ and $sum_1^infty$$a_n$
converges.



Give
counterexamples to the following claims where $b_n$ =
$a_{n+1}$/$a_n$



(a) $a_n ≤ 1$ for all $n ≥ 1$.



(b) The sequence ${a_n}$ is non-increasing.



(c) $lim_{nto infty}b_n$ exists.



(d) If $lim_{nto infty}b_n$ exists, then $lim_{nto infty}b_n < 1$.



(e) The sequence ${b_n}$ is bounded.



(f) If $limsup_{nto infty}b_n$ exists, then $limsup_{nto infty}b_nleq 1$.




My attempt: $2/n^3$ works for (a) and (d). I would really appreciate help for the other counter-examples.






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 11 '18 at 7:03









Robert Z

101k1070143




101k1070143










asked Dec 11 '18 at 6:25









childishsadbinochildishsadbino

1148




1148












  • $begingroup$
    I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:38










  • $begingroup$
    You should add these to your post.
    $endgroup$
    – xbh
    Dec 11 '18 at 6:44










  • $begingroup$
    $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:47










  • $begingroup$
    @mathworker21 no, we have to come up with counter-examples for these statements.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:50










  • $begingroup$
    @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:51




















  • $begingroup$
    I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:38










  • $begingroup$
    You should add these to your post.
    $endgroup$
    – xbh
    Dec 11 '18 at 6:44










  • $begingroup$
    $a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:47










  • $begingroup$
    @mathworker21 no, we have to come up with counter-examples for these statements.
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 6:50










  • $begingroup$
    @childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
    $endgroup$
    – mathworker21
    Dec 11 '18 at 6:51


















$begingroup$
I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
$endgroup$
– childishsadbino
Dec 11 '18 at 6:38




$begingroup$
I came up with an answer for part (b) as well, so I'm now just looking for help on (c), (e), and (f). My proposed solution isn't a counterexample for these cases, I believe.
$endgroup$
– childishsadbino
Dec 11 '18 at 6:38












$begingroup$
You should add these to your post.
$endgroup$
– xbh
Dec 11 '18 at 6:44




$begingroup$
You should add these to your post.
$endgroup$
– xbh
Dec 11 '18 at 6:44












$begingroup$
$a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
$endgroup$
– mathworker21
Dec 11 '18 at 6:47




$begingroup$
$a_n := frac{1}{2^n}$ satisfies $lim_n b_n = frac{1}{2}$, so (c),(e),(f) are satisfied
$endgroup$
– mathworker21
Dec 11 '18 at 6:47












$begingroup$
@mathworker21 no, we have to come up with counter-examples for these statements.
$endgroup$
– childishsadbino
Dec 11 '18 at 6:50




$begingroup$
@mathworker21 no, we have to come up with counter-examples for these statements.
$endgroup$
– childishsadbino
Dec 11 '18 at 6:50












$begingroup$
@childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
$endgroup$
– mathworker21
Dec 11 '18 at 6:51






$begingroup$
@childishsadbino my bad. idk, just have $a_n$ alternate between $frac{1}{2^n}$ and $frac{1}{n^2}$, or something
$endgroup$
– mathworker21
Dec 11 '18 at 6:51












2 Answers
2






active

oldest

votes


















1












$begingroup$

Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    $begingroup$
    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    $endgroup$
    – Robert Z
    Dec 11 '18 at 7:17












  • $begingroup$
    Makes sense now! Thank you so much!
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:21



















1












$begingroup$

For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This should serve as a counter-example for (e) as well, right?
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:02










  • $begingroup$
    Yeah.${{{{{{}}}}}}$
    $endgroup$
    – xbh
    Dec 11 '18 at 7:02












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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034961%2fcounterexamples-related-to-a-convergent-positive-series%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    $begingroup$
    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    $endgroup$
    – Robert Z
    Dec 11 '18 at 7:17












  • $begingroup$
    Makes sense now! Thank you so much!
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:21
















1












$begingroup$

Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    $begingroup$
    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    $endgroup$
    – Robert Z
    Dec 11 '18 at 7:17












  • $begingroup$
    Makes sense now! Thank you so much!
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:21














1












1








1





$begingroup$

Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?






share|cite|improve this answer











$endgroup$



Hint. As regards (c) and (e) consider
$$a_n=frac{1+(-1)^n+2^{-n}}{n^2}.$$
Can you modify it in order to obtain a counterexample for (f)?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 11 '18 at 6:58

























answered Dec 11 '18 at 6:52









Robert ZRobert Z

101k1070143




101k1070143












  • $begingroup$
    I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    $begingroup$
    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    $endgroup$
    – Robert Z
    Dec 11 '18 at 7:17












  • $begingroup$
    Makes sense now! Thank you so much!
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:21


















  • $begingroup$
    I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:14






  • 1




    $begingroup$
    @childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
    $endgroup$
    – Robert Z
    Dec 11 '18 at 7:17












  • $begingroup$
    Makes sense now! Thank you so much!
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:21
















$begingroup$
I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
$endgroup$
– childishsadbino
Dec 11 '18 at 7:14




$begingroup$
I apologize, but I do not immediately see how I can modify your example to get a counterexample for (f).
$endgroup$
– childishsadbino
Dec 11 '18 at 7:14




1




1




$begingroup$
@childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
$endgroup$
– Robert Z
Dec 11 '18 at 7:17






$begingroup$
@childishsadbino For example replace the numerator of the given $a_n$ with $2+(-1)^n$
$endgroup$
– Robert Z
Dec 11 '18 at 7:17














$begingroup$
Makes sense now! Thank you so much!
$endgroup$
– childishsadbino
Dec 11 '18 at 7:21




$begingroup$
Makes sense now! Thank you so much!
$endgroup$
– childishsadbino
Dec 11 '18 at 7:21











1












$begingroup$

For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This should serve as a counter-example for (e) as well, right?
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:02










  • $begingroup$
    Yeah.${{{{{{}}}}}}$
    $endgroup$
    – xbh
    Dec 11 '18 at 7:02
















1












$begingroup$

For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This should serve as a counter-example for (e) as well, right?
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:02










  • $begingroup$
    Yeah.${{{{{{}}}}}}$
    $endgroup$
    – xbh
    Dec 11 '18 at 7:02














1












1








1





$begingroup$

For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$






share|cite|improve this answer









$endgroup$



For (c), consider "merging" two convergent series:
$$
a_{2n} = frac 1{3^n}, a_{2n-1}= frac 1{2^n},
$$
then
$$
varlimsup b_n = +infty, varliminf b_n = 0.
$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 11 '18 at 6:59









xbhxbh

6,3201522




6,3201522












  • $begingroup$
    This should serve as a counter-example for (e) as well, right?
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:02










  • $begingroup$
    Yeah.${{{{{{}}}}}}$
    $endgroup$
    – xbh
    Dec 11 '18 at 7:02


















  • $begingroup$
    This should serve as a counter-example for (e) as well, right?
    $endgroup$
    – childishsadbino
    Dec 11 '18 at 7:02










  • $begingroup$
    Yeah.${{{{{{}}}}}}$
    $endgroup$
    – xbh
    Dec 11 '18 at 7:02
















$begingroup$
This should serve as a counter-example for (e) as well, right?
$endgroup$
– childishsadbino
Dec 11 '18 at 7:02




$begingroup$
This should serve as a counter-example for (e) as well, right?
$endgroup$
– childishsadbino
Dec 11 '18 at 7:02












$begingroup$
Yeah.${{{{{{}}}}}}$
$endgroup$
– xbh
Dec 11 '18 at 7:02




$begingroup$
Yeah.${{{{{{}}}}}}$
$endgroup$
– xbh
Dec 11 '18 at 7:02


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034961%2fcounterexamples-related-to-a-convergent-positive-series%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
Read more