Testing convergence of improper integral with a variable a
$begingroup$
I have trouble determining for which $ain mathbb{R};$ the following improper integral converges:
$$int_0^1frac{ln(x)}{x^a}dx$$
I have tried the following:
$left|frac{ln(x)}{x^a}right|=frac{-ln(x)}{x^a}$ for $0<xleq 1$
and:
$frac{1}{x}>-ln(x)$ on this interval.
So $frac{frac{1}{x}}{x^a}>frac{-ln(x)}{x^a}$ (on this interval)
$frac{frac{1}{x}}{x^a}$ can be written as a p-integral:
$int_0^1 x^{-p} dx$ with $p = a+1.$
Thus, I concluded, the improper integral converges for $a<0$ since the p-integral does too.
However, I'm asking myself whether this is the right way to tackle these kinds of problems, especially since I have no idea how to prove/disprove convergence for $ageq 0.$
Is what I have done above right? And how should I go about testing convergence for $ageq 0$?
convergence improper-integrals
edited Dec 1 '18 at 20:17
user376343
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
$endgroup$
add a comment |
$begingroup$
I have trouble determining for which $ain mathbb{R};$ the following improper integral converges:
$$int_0^1frac{ln(x)}{x^a}dx$$
I have tried the following:
$left|frac{ln(x)}{x^a}right|=frac{-ln(x)}{x^a}$ for $0<xleq 1$
and:
$frac{1}{x}>-ln(x)$ on this interval.
So $frac{frac{1}{x}}{x^a}>frac{-ln(x)}{x^a}$ (on this interval)
$frac{frac{1}{x}}{x^a}$ can be written as a p-integral:
$int_0^1 x^{-p} dx$ with $p = a+1.$
Thus, I concluded, the improper integral converges for $a<0$ since the p-integral does too.
However, I'm asking myself whether this is the right way to tackle these kinds of problems, especially since I have no idea how to prove/disprove convergence for $ageq 0.$
Is what I have done above right? And how should I go about testing convergence for $ageq 0$?
convergence improper-integrals
edited Dec 1 '18 at 20:17
user376343
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
$endgroup$
add a comment |
$begingroup$
I have trouble determining for which $ain mathbb{R};$ the following improper integral converges:
$$int_0^1frac{ln(x)}{x^a}dx$$
I have tried the following:
$left|frac{ln(x)}{x^a}right|=frac{-ln(x)}{x^a}$ for $0<xleq 1$
and:
$frac{1}{x}>-ln(x)$ on this interval.
So $frac{frac{1}{x}}{x^a}>frac{-ln(x)}{x^a}$ (on this interval)
$frac{frac{1}{x}}{x^a}$ can be written as a p-integral:
$int_0^1 x^{-p} dx$ with $p = a+1.$
Thus, I concluded, the improper integral converges for $a<0$ since the p-integral does too.
However, I'm asking myself whether this is the right way to tackle these kinds of problems, especially since I have no idea how to prove/disprove convergence for $ageq 0.$
Is what I have done above right? And how should I go about testing convergence for $ageq 0$?
convergence improper-integrals
edited Dec 1 '18 at 20:17
user376343
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
$endgroup$
I have trouble determining for which $ain mathbb{R};$ the following improper integral converges:
$$int_0^1frac{ln(x)}{x^a}dx$$
I have tried the following:
$left|frac{ln(x)}{x^a}right|=frac{-ln(x)}{x^a}$ for $0<xleq 1$
and:
$frac{1}{x}>-ln(x)$ on this interval.
So $frac{frac{1}{x}}{x^a}>frac{-ln(x)}{x^a}$ (on this interval)
$frac{frac{1}{x}}{x^a}$ can be written as a p-integral:
$int_0^1 x^{-p} dx$ with $p = a+1.$
Thus, I concluded, the improper integral converges for $a<0$ since the p-integral does too.
However, I'm asking myself whether this is the right way to tackle these kinds of problems, especially since I have no idea how to prove/disprove convergence for $ageq 0.$
Is what I have done above right? And how should I go about testing convergence for $ageq 0$?
convergence improper-integrals
convergence improper-integrals
edited Dec 1 '18 at 20:17
user376343
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
edited Dec 1 '18 at 20:17
user376343
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
edited Dec 1 '18 at 20:17
user376343
3,7883828
edited Dec 1 '18 at 20:17
user376343
3,7883828
edited Dec 1 '18 at 20:17
user376343
3,7883828
3,7883828
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
asked Dec 1 '18 at 19:55
Renze SutersRenze Suters
11
11
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%2f3021746%2ftesting-convergence-of-improper-integral-with-a-variable-a%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%2f3021746%2ftesting-convergence-of-improper-integral-with-a-variable-a%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
