Prove sequence diverges to infinity

[sitia]

Homework Statement

I have to prove that the sequence a(n)=(n^3-n +1)/(2n+4) diverges to infinity.

Homework Equations

The Attempt at a Solution

Observe that n^3-n +1 > (1/2)n^3 and 2n+4≤4n in n≥2

I am now stuck on how to proceed. I am confused on opposite inequalities for the numerator and denom. Can you direct as to how I'm to proceed?

Thanks!

 

[Curious3141]

Homework Statement

I have to prove that the sequence a(n)=(n^3-n +1)/(2n+4) diverges to infinity.

Homework Equations

The Attempt at a Solution

Observe that n^3-n +1 > (1/2)n^3 and 2n+4≤4n in n≥2

I am now stuck on how to proceed. I am confused on opposite inequalities for the numerator and denom. Can you direct as to how I'm to proceed?

Thanks!

To "see" it, just observe that for large n, the numerator approaches n³, while the denominator approaches 2n.

To prove it, just do the long division to get a quadratic quotient (and a remainder, which vanishes at the limit). Or use synthetic division. From this point on, the limit should be obvious (although you can complete the square for the quadratic to make it even more rigorous).