how to prove that $f(n)=n^3+nlog^2n$ = $theta(n^3)$?












0














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










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  • 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


















0














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










share|cite|improve this question
























  • 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
















0












0








0







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










share|cite|improve this question















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






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edited Nov 20 at 18:25









Key Flex

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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




















  • 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












1 Answer
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Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$






share|cite|improve this answer





















  • 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













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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0














Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$






share|cite|improve this answer





















  • 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


















0














Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$






share|cite|improve this answer





















  • 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
















0












0








0






Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$






share|cite|improve this answer












Hint: $lim_{n to infty} frac{log n}{sqrt n} = (infty / infty) = (by L'Hospital) lim_{n to infty}frac{2sqrt n}{n} =0$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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




















  • 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




















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