Prove $sum a_n b_n$ diverges if $a_n$ diverges, $a_n>0$, and $liminf_n b_n >0$
$begingroup$
I'm having trouble starting this proof. From the initial hypothesis, we know for some $n>N$, $b_n>0$. Since $a_n>0$ and $a_n$ diverges, for some $n>N'$, $a_ngeq varepsilon$. Then $a_n b_n>0$, so if $sum a_n b_n$ is a series of positive numbers. I'm not sure how to show it diverges however, so any help would be appreciated!
real-analysis sequences-and-series convergence
asked Dec 3 '18 at 2:51
t.perezt.perez
619
$endgroup$
add a comment |
$begingroup$
I'm having trouble starting this proof. From the initial hypothesis, we know for some $n>N$, $b_n>0$. Since $a_n>0$ and $a_n$ diverges, for some $n>N'$, $a_ngeq varepsilon$. Then $a_n b_n>0$, so if $sum a_n b_n$ is a series of positive numbers. I'm not sure how to show it diverges however, so any help would be appreciated!
real-analysis sequences-and-series convergence
asked Dec 3 '18 at 2:51
t.perezt.perez
619
$endgroup$
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03
add a comment |
$begingroup$
I'm having trouble starting this proof. From the initial hypothesis, we know for some $n>N$, $b_n>0$. Since $a_n>0$ and $a_n$ diverges, for some $n>N'$, $a_ngeq varepsilon$. Then $a_n b_n>0$, so if $sum a_n b_n$ is a series of positive numbers. I'm not sure how to show it diverges however, so any help would be appreciated!
real-analysis sequences-and-series convergence
asked Dec 3 '18 at 2:51
t.perezt.perez
619
$endgroup$
I'm having trouble starting this proof. From the initial hypothesis, we know for some $n>N$, $b_n>0$. Since $a_n>0$ and $a_n$ diverges, for some $n>N'$, $a_ngeq varepsilon$. Then $a_n b_n>0$, so if $sum a_n b_n$ is a series of positive numbers. I'm not sure how to show it diverges however, so any help would be appreciated!
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
asked Dec 3 '18 at 2:51
t.perezt.perez
619
asked Dec 3 '18 at 2:51
t.perezt.perez
619
asked Dec 3 '18 at 2:51
t.perezt.perez
619
asked Dec 3 '18 at 2:51
t.perezt.perez
619
asked Dec 3 '18 at 2:51
t.perezt.perez
619
619
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03
add a comment |
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let
$$
liminf_{nto infty}b_n=2delta>0
$$
Then, for large enough $N$, $b_ngeq delta$ for all $ngeq N$. Then,
$$
sum_{n=N}^infty b_na_ngeq deltasum_{n=N}^infty a_n=+infty
$$
since $lim_{nto infty}a_nne 0$.
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
$endgroup$
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
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%2f3023548%2fprove-sum-a-n-b-n-diverges-if-a-n-diverges-a-n0-and-lim-inf-n-b-n-0%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
$begingroup$
Let
$$
liminf_{nto infty}b_n=2delta>0
$$
Then, for large enough $N$, $b_ngeq delta$ for all $ngeq N$. Then,
$$
sum_{n=N}^infty b_na_ngeq deltasum_{n=N}^infty a_n=+infty
$$
since $lim_{nto infty}a_nne 0$.
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
$endgroup$
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
add a comment |
$begingroup$
Let
$$
liminf_{nto infty}b_n=2delta>0
$$
Then, for large enough $N$, $b_ngeq delta$ for all $ngeq N$. Then,
$$
sum_{n=N}^infty b_na_ngeq deltasum_{n=N}^infty a_n=+infty
$$
since $lim_{nto infty}a_nne 0$.
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
$endgroup$
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
add a comment |
$begingroup$
Let
$$
liminf_{nto infty}b_n=2delta>0
$$
Then, for large enough $N$, $b_ngeq delta$ for all $ngeq N$. Then,
$$
sum_{n=N}^infty b_na_ngeq deltasum_{n=N}^infty a_n=+infty
$$
since $lim_{nto infty}a_nne 0$.
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
$endgroup$
Let
$$
liminf_{nto infty}b_n=2delta>0
$$
Then, for large enough $N$, $b_ngeq delta$ for all $ngeq N$. Then,
$$
sum_{n=N}^infty b_na_ngeq deltasum_{n=N}^infty a_n=+infty
$$
since $lim_{nto infty}a_nne 0$.
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
edited Dec 3 '18 at 4:08
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
answered Dec 3 '18 at 3:00
qbertqbert
22.1k32561
22.1k32561
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
add a comment |
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
Thank you! My only question is, where does the 2 go in the $2delta$ term?
$endgroup$
– t.perez
Dec 5 '18 at 2:12
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
$begingroup$
okay, that makes sense now. thank you!
$endgroup$
– t.perez
Dec 5 '18 at 2:24
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%2f3023548%2fprove-sum-a-n-b-n-diverges-if-a-n-diverges-a-n0-and-lim-inf-n-b-n-0%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
$begingroup$
@qbert isn't the answer below correct regardless?
$endgroup$
– clark
Dec 3 '18 at 3:03