Abel-Dini-Type Theorem for convergent seriesContest problem about convergent seriesA generalization of Cauchy's condensation testA convergent series propertyConvergence of the series $sumlimits_ne^-a_n$Improving a proof for a basic property of convergent seriesConvergent series with conditionSize of a convergent series restricted to primesCan the speed of divergence/convergence of a series be controlled arbitrarily?Limit of the series $lim_nrightarrow inftyfrac1s_nsum_k=1^na_kx_k$Absolutely convergent series of complex functions.
Multi tool use
Does an advisor owe his/her student anything? Will an advisor keep a PhD student only out of pity?
Does the Linux kernel need a file system to run?
How to get directions in deep space?
Make a Bowl of Alphabet Soup
The Digit Triangles
How to convince somebody that he is fit for something else, but not this job?
Has the laser at Magurele, Romania reached a tenth of the Sun's power?
Why is the Sun approximated as a black body at ~ 5800 K?
A variation to the phrase "hanging over my shoulders"
A Trivial Diagnosis
Do we have to expect a queue for the shuttle from Watford Junction to Harry Potter Studio?
Multiplicative persistence
Change the color of a single dot in `ddot` symbol
What fields between the rationals and the reals allow a good notion of 2D distance?
How to make money from a browser who sees 5 seconds into the future of any web page?
Microchip documentation does not label CAN buss pins on micro controller pinout diagram
Can I say "fingers" when referring to toes?
Does the reader need to like the PoV character?
What does Apple's new App Store requirement mean
Is there any evidence that Cleopatra and Caesarion considered fleeing to India to escape the Romans?
Biological Blimps: Propulsion
Why is it that I can sometimes guess the next note?
Strong empirical falsification of quantum mechanics based on vacuum energy density?
Quoting Keynes in a lecture
Abel-Dini-Type Theorem for convergent series
Contest problem about convergent seriesA generalization of Cauchy's condensation testA convergent series propertyConvergence of the series $sumlimits_ne^-a_n$Improving a proof for a basic property of convergent seriesConvergent series with conditionSize of a convergent series restricted to primesCan the speed of divergence/convergence of a series be controlled arbitrarily?Limit of the series $lim_nrightarrow inftyfrac1s_nsum_k=1^na_kx_k$Absolutely convergent series of complex functions.
$begingroup$
Given are a continuous, strictly increasing and convex function $varphi:[0,infty)to[0,infty)$ and a sequence $(x_j)$ of positive real numbers such that
beginalign*
sum_j=1^inftyvarphi(x_j)
endalign*
converges.
Question: Can we always find a sequence $(lambda _j)$ of positive real numbers which decreases to 0 and so that
beginalign*
sum_j=1^inftyvarphi(x_j/lambda_j)
endalign*
still converges?
This is true for linear $varphi$ by the Abel-Dini-Theorem, and in this case one can take $lambda_j=sqrtx_j+1+x_j+2+ldots$. But what about the general case? Any help is highly appreciated. Thanks in advance!
sequences-and-series convergence
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
$endgroup$
add a comment |
$begingroup$
Given are a continuous, strictly increasing and convex function $varphi:[0,infty)to[0,infty)$ and a sequence $(x_j)$ of positive real numbers such that
beginalign*
sum_j=1^inftyvarphi(x_j)
endalign*
converges.
Question: Can we always find a sequence $(lambda _j)$ of positive real numbers which decreases to 0 and so that
beginalign*
sum_j=1^inftyvarphi(x_j/lambda_j)
endalign*
still converges?
This is true for linear $varphi$ by the Abel-Dini-Theorem, and in this case one can take $lambda_j=sqrtx_j+1+x_j+2+ldots$. But what about the general case? Any help is highly appreciated. Thanks in advance!
sequences-and-series convergence
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
$endgroup$
add a comment |
$begingroup$
Given are a continuous, strictly increasing and convex function $varphi:[0,infty)to[0,infty)$ and a sequence $(x_j)$ of positive real numbers such that
beginalign*
sum_j=1^inftyvarphi(x_j)
endalign*
converges.
Question: Can we always find a sequence $(lambda _j)$ of positive real numbers which decreases to 0 and so that
beginalign*
sum_j=1^inftyvarphi(x_j/lambda_j)
endalign*
still converges?
This is true for linear $varphi$ by the Abel-Dini-Theorem, and in this case one can take $lambda_j=sqrtx_j+1+x_j+2+ldots$. But what about the general case? Any help is highly appreciated. Thanks in advance!
sequences-and-series convergence
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
$endgroup$
Given are a continuous, strictly increasing and convex function $varphi:[0,infty)to[0,infty)$ and a sequence $(x_j)$ of positive real numbers such that
beginalign*
sum_j=1^inftyvarphi(x_j)
endalign*
converges.
Question: Can we always find a sequence $(lambda _j)$ of positive real numbers which decreases to 0 and so that
beginalign*
sum_j=1^inftyvarphi(x_j/lambda_j)
endalign*
still converges?
This is true for linear $varphi$ by the Abel-Dini-Theorem, and in this case one can take $lambda_j=sqrtx_j+1+x_j+2+ldots$. But what about the general case? Any help is highly appreciated. Thanks in advance!
sequences-and-series convergence
sequences-and-series convergence
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
asked Mar 14 at 16:28
sranthropsranthrop
7,1111925
7,1111925
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No - here is a counterexample. Define
$$phi(x)=begincases
0&x=0\
exp(-1/x)&0<xleq 1/2\
exp(-2)+4exp(-2)(x-tfrac12)&xgeq 1/2.
endcases$$
For $0<x<1/2$ we have $phi'(x)=x^-2exp(-1/x)$ and $phi''(x)=(x^-4-2x^-3)exp(-1/x)>0$ which ensures that $phi$ is continuous, strictly increasing and convex.
For $0<lambdaleq 1$ and $0<y^lambda<exp(-2)$ we have
$$phi(tfrac1lambda phi^-1(x))=phi(-1/lambdalog(y))=y^lambda.$$
For an extreme example take $x_j=phi^-1(1/j(log j)^2)$ for $jgeq 10$ (and $x_1,dots,x_9=1$ say). Then
$$sum_j=10^infty phi(x_j/lambda)=sum_j=10^infty frac1j^lambda(log j)^2lambda$$
which converges for $lambda=1$ but not for any $lambda<1.$
answered Mar 14 at 18:30
DapDap
18.9k842
$endgroup$
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
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%2f3148216%2fabel-dini-type-theorem-for-convergent-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
$begingroup$
No - here is a counterexample. Define
$$phi(x)=begincases
0&x=0\
exp(-1/x)&0<xleq 1/2\
exp(-2)+4exp(-2)(x-tfrac12)&xgeq 1/2.
endcases$$
For $0<x<1/2$ we have $phi'(x)=x^-2exp(-1/x)$ and $phi''(x)=(x^-4-2x^-3)exp(-1/x)>0$ which ensures that $phi$ is continuous, strictly increasing and convex.
For $0<lambdaleq 1$ and $0<y^lambda<exp(-2)$ we have
$$phi(tfrac1lambda phi^-1(x))=phi(-1/lambdalog(y))=y^lambda.$$
For an extreme example take $x_j=phi^-1(1/j(log j)^2)$ for $jgeq 10$ (and $x_1,dots,x_9=1$ say). Then
$$sum_j=10^infty phi(x_j/lambda)=sum_j=10^infty frac1j^lambda(log j)^2lambda$$
which converges for $lambda=1$ but not for any $lambda<1.$
answered Mar 14 at 18:30
DapDap
18.9k842
$endgroup$
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
add a comment |
$begingroup$
No - here is a counterexample. Define
$$phi(x)=begincases
0&x=0\
exp(-1/x)&0<xleq 1/2\
exp(-2)+4exp(-2)(x-tfrac12)&xgeq 1/2.
endcases$$
For $0<x<1/2$ we have $phi'(x)=x^-2exp(-1/x)$ and $phi''(x)=(x^-4-2x^-3)exp(-1/x)>0$ which ensures that $phi$ is continuous, strictly increasing and convex.
For $0<lambdaleq 1$ and $0<y^lambda<exp(-2)$ we have
$$phi(tfrac1lambda phi^-1(x))=phi(-1/lambdalog(y))=y^lambda.$$
For an extreme example take $x_j=phi^-1(1/j(log j)^2)$ for $jgeq 10$ (and $x_1,dots,x_9=1$ say). Then
$$sum_j=10^infty phi(x_j/lambda)=sum_j=10^infty frac1j^lambda(log j)^2lambda$$
which converges for $lambda=1$ but not for any $lambda<1.$
answered Mar 14 at 18:30
DapDap
18.9k842
$endgroup$
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
add a comment |
$begingroup$
No - here is a counterexample. Define
$$phi(x)=begincases
0&x=0\
exp(-1/x)&0<xleq 1/2\
exp(-2)+4exp(-2)(x-tfrac12)&xgeq 1/2.
endcases$$
For $0<x<1/2$ we have $phi'(x)=x^-2exp(-1/x)$ and $phi''(x)=(x^-4-2x^-3)exp(-1/x)>0$ which ensures that $phi$ is continuous, strictly increasing and convex.
For $0<lambdaleq 1$ and $0<y^lambda<exp(-2)$ we have
$$phi(tfrac1lambda phi^-1(x))=phi(-1/lambdalog(y))=y^lambda.$$
For an extreme example take $x_j=phi^-1(1/j(log j)^2)$ for $jgeq 10$ (and $x_1,dots,x_9=1$ say). Then
$$sum_j=10^infty phi(x_j/lambda)=sum_j=10^infty frac1j^lambda(log j)^2lambda$$
which converges for $lambda=1$ but not for any $lambda<1.$
answered Mar 14 at 18:30
DapDap
18.9k842
$endgroup$
No - here is a counterexample. Define
$$phi(x)=begincases
0&x=0\
exp(-1/x)&0<xleq 1/2\
exp(-2)+4exp(-2)(x-tfrac12)&xgeq 1/2.
endcases$$
For $0<x<1/2$ we have $phi'(x)=x^-2exp(-1/x)$ and $phi''(x)=(x^-4-2x^-3)exp(-1/x)>0$ which ensures that $phi$ is continuous, strictly increasing and convex.
For $0<lambdaleq 1$ and $0<y^lambda<exp(-2)$ we have
$$phi(tfrac1lambda phi^-1(x))=phi(-1/lambdalog(y))=y^lambda.$$
For an extreme example take $x_j=phi^-1(1/j(log j)^2)$ for $jgeq 10$ (and $x_1,dots,x_9=1$ say). Then
$$sum_j=10^infty phi(x_j/lambda)=sum_j=10^infty frac1j^lambda(log j)^2lambda$$
which converges for $lambda=1$ but not for any $lambda<1.$
answered Mar 14 at 18:30
DapDap
18.9k842
answered Mar 14 at 18:30
DapDap
18.9k842
answered Mar 14 at 18:30
DapDap
18.9k842
answered Mar 14 at 18:30
DapDap
18.9k842
18.9k842
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
add a comment |
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
$begingroup$
Great, thank you!
$endgroup$
– sranthrop
Mar 14 at 19:15
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%2f3148216%2fabel-dini-type-theorem-for-convergent-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
6T 9ljJ9
