Applications/examples of these properties?
$begingroup$
Here are two interesting properties on series :
The first one :
Let $(u_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} u_n=+infty$. Then there exists $(v_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} v_n=+infty$ with $v_n underset{nto +infty}{=} o(u_n)$.
Then the second one :
If $(g_k)_{kin mathbb{N}^*}$ is a strictly increasing sequence of strictly positive integers such that there exists $nu >0$ with for all $kin mathbb{N}^*$, $(g_{k+1}-g_k)le nu (g_k-g_{k-1})$ and if $(a_n)in (mathbb{R^+})^{mathbb{N}}$ is strictly decreasing then $sum limits_{nge 1}a_n<+infty$ iff $sum limits_{kge 1}(g_{k+1}-g_k)a_{g_k}<+infty$.
Could we find applications or examples of these two properties ?
Thanks in advance !
sequences-and-series limits divergent-series
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
$endgroup$
add a comment |
$begingroup$
Here are two interesting properties on series :
The first one :
Let $(u_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} u_n=+infty$. Then there exists $(v_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} v_n=+infty$ with $v_n underset{nto +infty}{=} o(u_n)$.
Then the second one :
If $(g_k)_{kin mathbb{N}^*}$ is a strictly increasing sequence of strictly positive integers such that there exists $nu >0$ with for all $kin mathbb{N}^*$, $(g_{k+1}-g_k)le nu (g_k-g_{k-1})$ and if $(a_n)in (mathbb{R^+})^{mathbb{N}}$ is strictly decreasing then $sum limits_{nge 1}a_n<+infty$ iff $sum limits_{kge 1}(g_{k+1}-g_k)a_{g_k}<+infty$.
Could we find applications or examples of these two properties ?
Thanks in advance !
sequences-and-series limits divergent-series
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
$endgroup$
add a comment |
$begingroup$
Here are two interesting properties on series :
The first one :
Let $(u_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} u_n=+infty$. Then there exists $(v_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} v_n=+infty$ with $v_n underset{nto +infty}{=} o(u_n)$.
Then the second one :
If $(g_k)_{kin mathbb{N}^*}$ is a strictly increasing sequence of strictly positive integers such that there exists $nu >0$ with for all $kin mathbb{N}^*$, $(g_{k+1}-g_k)le nu (g_k-g_{k-1})$ and if $(a_n)in (mathbb{R^+})^{mathbb{N}}$ is strictly decreasing then $sum limits_{nge 1}a_n<+infty$ iff $sum limits_{kge 1}(g_{k+1}-g_k)a_{g_k}<+infty$.
Could we find applications or examples of these two properties ?
Thanks in advance !
sequences-and-series limits divergent-series
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
$endgroup$
Here are two interesting properties on series :
The first one :
Let $(u_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} u_n=+infty$. Then there exists $(v_n)in(mathbb{R^+})^{mathbb{N}}$ such that $sum limits_{nge 0} v_n=+infty$ with $v_n underset{nto +infty}{=} o(u_n)$.
Then the second one :
If $(g_k)_{kin mathbb{N}^*}$ is a strictly increasing sequence of strictly positive integers such that there exists $nu >0$ with for all $kin mathbb{N}^*$, $(g_{k+1}-g_k)le nu (g_k-g_{k-1})$ and if $(a_n)in (mathbb{R^+})^{mathbb{N}}$ is strictly decreasing then $sum limits_{nge 1}a_n<+infty$ iff $sum limits_{kge 1}(g_{k+1}-g_k)a_{g_k}<+infty$.
Could we find applications or examples of these two properties ?
Thanks in advance !
sequences-and-series limits divergent-series
sequences-and-series limits divergent-series
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
edited Nov 29 '18 at 20:35
Maman
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
asked Nov 28 '18 at 0:50
MamanMaman
1,174722
1,174722
add a comment |
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%2f3016549%2fapplications-examples-of-these-properties%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%2f3016549%2fapplications-examples-of-these-properties%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
