# How to find the standard theta notation of this?

Hi i am practising standard theta notation:

How could i find the standard theta notation of the following :

• 2n + 3n^2(log n)^3 + 2 and
• ((2n^2 + 7n+5)(7n+1))/7

So far i have used the "drop constants and find dominating term" strategy where we remove the constants from the expressions and then we find the biggest and dominating value and i got:

• Θ(n^2(logn)^3) and
• Θ(n^2) respectively, however i am not sure if this is correct, could someone please help me out. Thank you.
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Feb 5 at 0:52
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Feb 13 at 0:06
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Perform this test:

• does $$\dfrac{2n + 3n^2\log^3 n + 2}{3n^2\log^3n}$$ tend to a constant ?

• does $$\dfrac{(2n^2 + 7n+5)(7n+1)}{n^2}$$ tend to a constant ?

• i dont understand please elaborate
– zeek
Feb 5 at 11:08
• I hate to nitpick, but 2+sin(n) = Θ(1), yet (2+sin(n))/1 does not tend to a constant.
– Stef
Feb 5 at 11:26
• @Stef: tending to a constant is a sufficient condition. I knew that someone would nitpick. Feb 5 at 14:12
• First Expression: $$2n + 3n^2(log n)^3 + 2$$ is indeed correct.
• Second Expression: $$\dfrac{(2n^2 + 7n+5)(7n+1)}{7}$$ is not correct. The leading expression of the second expression is $$\dfrac{14}{7}n^3$$ and not $$2n^2$$. It seems that you missed multiplying the two expressions by each other. Which give you the following: $$\dfrac{14n^3 + 51n^2 + 42n + 5}{7}$$. Thus, the leading expression is $$\dfrac{14n^3}{7}=2n^3$$.

Thus, the second expression would give you that it is $$\Theta(2n^3)$$.

I'd like to advise you to read more about the formal definition of the function $$\Theta$$, which requires you to know about the formal definitions of both $$O$$ and $$\Omega$$. In the expressions you gave, proving $$\Theta(g(n))$$ is easy. However, in many other expressions and algorithms , proving that $$f(n) = \Theta(g(n))$$ isn't easy. It would require you to prove both $$f(n) = O(g(n))$$ and $$f(n) = \Omega(g(n))$$.