how to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$?
i have $f(n)=n^3+nlog^2n$
and i was trieng to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$. but i feel that i am doing it all wrong , which means i understand why the statent is true but i don't know if the way i proved it was enough..
here is what i did so far :
first i need to prove that :
$f(n)=n^3+nlog^2n$ = $O(n^3)$
We can choose constant C like this : if we devide the equation with $n^3$ then we get : $1+nlog^2n/n^3$ $<=$ $C $ but $nlog^2n/n^3$ <= 1 for every n so if we choose C to be C>=2 then this will be proved.
second i need to prove :$f(n)=n^3+nlog^2n$ = $pi(n^3)$
this i think doesn't need to be proved because if i choose any C<1 then for every n we will get : $n^3+nlog^2n$ >= $C(n^3)$
here is
algorithms asymptotics
add a comment |
i have $f(n)=n^3+nlog^2n$
and i was trieng to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$. but i feel that i am doing it all wrong , which means i understand why the statent is true but i don't know if the way i proved it was enough..
here is what i did so far :
first i need to prove that :
$f(n)=n^3+nlog^2n$ = $O(n^3)$
We can choose constant C like this : if we devide the equation with $n^3$ then we get : $1+nlog^2n/n^3$ $<=$ $C $ but $nlog^2n/n^3$ <= 1 for every n so if we choose C to be C>=2 then this will be proved.
second i need to prove :$f(n)=n^3+nlog^2n$ = $pi(n^3)$
this i think doesn't need to be proved because if i choose any C<1 then for every n we will get : $n^3+nlog^2n$ >= $C(n^3)$
here is
algorithms asymptotics
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03
add a comment |
i have $f(n)=n^3+nlog^2n$
and i was trieng to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$. but i feel that i am doing it all wrong , which means i understand why the statent is true but i don't know if the way i proved it was enough..
here is what i did so far :
first i need to prove that :
$f(n)=n^3+nlog^2n$ = $O(n^3)$
We can choose constant C like this : if we devide the equation with $n^3$ then we get : $1+nlog^2n/n^3$ $<=$ $C $ but $nlog^2n/n^3$ <= 1 for every n so if we choose C to be C>=2 then this will be proved.
second i need to prove :$f(n)=n^3+nlog^2n$ = $pi(n^3)$
this i think doesn't need to be proved because if i choose any C<1 then for every n we will get : $n^3+nlog^2n$ >= $C(n^3)$
here is
algorithms asymptotics
i have $f(n)=n^3+nlog^2n$
and i was trieng to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$. but i feel that i am doing it all wrong , which means i understand why the statent is true but i don't know if the way i proved it was enough..
here is what i did so far :
first i need to prove that :
$f(n)=n^3+nlog^2n$ = $O(n^3)$
We can choose constant C like this : if we devide the equation with $n^3$ then we get : $1+nlog^2n/n^3$ $<=$ $C $ but $nlog^2n/n^3$ <= 1 for every n so if we choose C to be C>=2 then this will be proved.
second i need to prove :$f(n)=n^3+nlog^2n$ = $pi(n^3)$
this i think doesn't need to be proved because if i choose any C<1 then for every n we will get : $n^3+nlog^2n$ >= $C(n^3)$
here is
algorithms asymptotics
algorithms asymptotics
edited Nov 20 at 18:25
Key Flex
7,52441232
7,52441232
asked Nov 20 at 18:18
jasmin
12
12
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03
add a comment |
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03
add a comment |
1 Answer
1
active
oldest
votes
Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
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%2f3006698%2fhow-to-prove-that-fn-n3n-log2n-thetan3%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
Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
add a comment |
Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
add a comment |
Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$
Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$
answered Nov 20 at 19:26
Vasya
3,3241515
3,3241515
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
add a comment |
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
i didn't really understand the hint .. is this a way toprove that nlog2n/n3 <= 1 ?
– jasmin
Nov 20 at 20:12
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
@jasmin: basically, $log n$ grows slower than $sqrt n$ so $log^2 n$ grows slower than $n$ which means $nlog^2 n/n^3< 1/n le 1$
– Vasya
Nov 20 at 20:35
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
To the proposer: For any positive $A,B$ we have $ lim_{xto infty}(ln x)^A/x^B=0.$
– DanielWainfleet
Nov 21 at 7:07
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3006698%2fhow-to-prove-that-fn-n3n-log2n-thetan3%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
What are $theta$ and pi?..... BTW the codes le and leq give $le$ while ge and geq give $ge$.
– DanielWainfleet
Nov 21 at 7:03