Mixing
What can we say about asymptotic behavior of primitive functions?
$begingroup$
Assume I am studying the series of the form $$sum_{k=1}^infty f(k)$$
where we assume $f$ to be some continuous monotonic function over the interval $langle1,+infty)$. We know that for these function we have this bound:
$$f(1)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(n)+int_1^nf(x)dx$$
or
$$f(n)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(1)+int_1^nf(x)dx$$
if $f$ is decreasing. My goal is to determine the behaviour of the partial sum $s_n=sum_{k=1}^nf(k)$. More precisely to find sequence $a_n$ such that $s_nsim a_n$ in the sense that $lim_{ntoinfty}frac{a_n}{s_n}=1.$My question is: Can i explicitly say that
$$s_nsimint_1^nf(k)$$
Assume there exists primitive $F$ to $f$ then rewriting the upper inequality (for increasing functions)
$$f(1)+F(n)-F(1)leq s_n leq f(n)+F(n)-F(1)$$
It seems that this should always hold but I can't seem to relate functions with their primitives. In fact $f(1), F(1)$ are some constants and $f(n)$ vanishes compared to $F(n)$ so $s_n$ should be bounded by $F(n)$ thus $s_nsim F(n)$. Is that true? If $f$ is a function and $F$ its primitive, what can we say about the limit
$$lim_{ntoinfty}frac{f(n)}{F(n)}$$
What about $$f(n)in O(F(n))$$
Does it hold?
real-analysis calculus sequences-and-series asymptotics
edited Jan 24 at 23:17
Bernard
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
$endgroup$
add a comment |
$begingroup$
Assume I am studying the series of the form $$sum_{k=1}^infty f(k)$$
where we assume $f$ to be some continuous monotonic function over the interval $langle1,+infty)$. We know that for these function we have this bound:
$$f(1)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(n)+int_1^nf(x)dx$$
or
$$f(n)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(1)+int_1^nf(x)dx$$
if $f$ is decreasing. My goal is to determine the behaviour of the partial sum $s_n=sum_{k=1}^nf(k)$. More precisely to find sequence $a_n$ such that $s_nsim a_n$ in the sense that $lim_{ntoinfty}frac{a_n}{s_n}=1.$My question is: Can i explicitly say that
$$s_nsimint_1^nf(k)$$
Assume there exists primitive $F$ to $f$ then rewriting the upper inequality (for increasing functions)
$$f(1)+F(n)-F(1)leq s_n leq f(n)+F(n)-F(1)$$
It seems that this should always hold but I can't seem to relate functions with their primitives. In fact $f(1), F(1)$ are some constants and $f(n)$ vanishes compared to $F(n)$ so $s_n$ should be bounded by $F(n)$ thus $s_nsim F(n)$. Is that true? If $f$ is a function and $F$ its primitive, what can we say about the limit
$$lim_{ntoinfty}frac{f(n)}{F(n)}$$
What about $$f(n)in O(F(n))$$
Does it hold?
real-analysis calculus sequences-and-series asymptotics
edited Jan 24 at 23:17
Bernard
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
$endgroup$
add a comment |
$begingroup$
Assume I am studying the series of the form $$sum_{k=1}^infty f(k)$$
where we assume $f$ to be some continuous monotonic function over the interval $langle1,+infty)$. We know that for these function we have this bound:
$$f(1)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(n)+int_1^nf(x)dx$$
or
$$f(n)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(1)+int_1^nf(x)dx$$
if $f$ is decreasing. My goal is to determine the behaviour of the partial sum $s_n=sum_{k=1}^nf(k)$. More precisely to find sequence $a_n$ such that $s_nsim a_n$ in the sense that $lim_{ntoinfty}frac{a_n}{s_n}=1.$My question is: Can i explicitly say that
$$s_nsimint_1^nf(k)$$
Assume there exists primitive $F$ to $f$ then rewriting the upper inequality (for increasing functions)
$$f(1)+F(n)-F(1)leq s_n leq f(n)+F(n)-F(1)$$
It seems that this should always hold but I can't seem to relate functions with their primitives. In fact $f(1), F(1)$ are some constants and $f(n)$ vanishes compared to $F(n)$ so $s_n$ should be bounded by $F(n)$ thus $s_nsim F(n)$. Is that true? If $f$ is a function and $F$ its primitive, what can we say about the limit
$$lim_{ntoinfty}frac{f(n)}{F(n)}$$
What about $$f(n)in O(F(n))$$
Does it hold?
real-analysis calculus sequences-and-series asymptotics
edited Jan 24 at 23:17
Bernard
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
$endgroup$
Assume I am studying the series of the form $$sum_{k=1}^infty f(k)$$
where we assume $f$ to be some continuous monotonic function over the interval $langle1,+infty)$. We know that for these function we have this bound:
$$f(1)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(n)+int_1^nf(x)dx$$
or
$$f(n)+int_1^nf(x)dxleqsum_{k=1}^nf(k)leq f(1)+int_1^nf(x)dx$$
if $f$ is decreasing. My goal is to determine the behaviour of the partial sum $s_n=sum_{k=1}^nf(k)$. More precisely to find sequence $a_n$ such that $s_nsim a_n$ in the sense that $lim_{ntoinfty}frac{a_n}{s_n}=1.$My question is: Can i explicitly say that
$$s_nsimint_1^nf(k)$$
Assume there exists primitive $F$ to $f$ then rewriting the upper inequality (for increasing functions)
$$f(1)+F(n)-F(1)leq s_n leq f(n)+F(n)-F(1)$$
It seems that this should always hold but I can't seem to relate functions with their primitives. In fact $f(1), F(1)$ are some constants and $f(n)$ vanishes compared to $F(n)$ so $s_n$ should be bounded by $F(n)$ thus $s_nsim F(n)$. Is that true? If $f$ is a function and $F$ its primitive, what can we say about the limit
$$lim_{ntoinfty}frac{f(n)}{F(n)}$$
What about $$f(n)in O(F(n))$$
Does it hold?
real-analysis calculus sequences-and-series asymptotics
real-analysis calculus sequences-and-series asymptotics
edited Jan 24 at 23:17
Bernard
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
edited Jan 24 at 23:17
Bernard
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
edited Jan 24 at 23:17
Bernard
123k741116
edited Jan 24 at 23:17
Bernard
123k741116
edited Jan 24 at 23:17
Bernard
123k741116
123k741116
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
asked Jan 24 at 23:13
Michal DvořákMichal Dvořák
1,012416
1,012416
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
$endgroup$
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
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%2f3086505%2fwhat-can-we-say-about-asymptotic-behavior-of-primitive-functions%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
$begingroup$
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
$endgroup$
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
add a comment |
$begingroup$
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
$endgroup$
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
add a comment |
$begingroup$
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
$endgroup$
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
answered Jan 25 at 4:24
Stefan LafonStefan Lafon
2,96019
2,96019
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
add a comment |
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
$begingroup$
Can we find an instance s.t. $F(n)in O(f(n))$ ?
$endgroup$
– Michal Dvořák
Jan 25 at 9:32
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%2f3086505%2fwhat-can-we-say-about-asymptotic-behavior-of-primitive-functions%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
