limit as n approaches infinity?

Question

Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit. If it diverges to infinity, state your answer as INF . If it diverges to negative infinity, state your answer as MINF . If it diverges without being infinity or negative infinity, state your answer as DIV .

limit as n approaches infinity
(((-1)^n)*n^3)/((n^3)+(5n^2)+9)

Answer 1

This sequence diverges. Considering just the fraction n³/(n³ + 5n² + 9), we see that the degree of the numerator equals the degree of the denominator, so it will converge to the ratio of the coefficients of the n³ terms, which is 1 (since both coefficients are 1). HOWEVER, there is also the (-1)^n term, which causes the function to change sign each term. Therefore, for very large even values of n, the function approaches 1, and for very large odd values of n, the function approaches -1. Because it keeps oscillating between these two values, the limit of the function does not exist.

Answer 2

It diverges but you don't know which way it goes, so DIV

Answer 3

diverges: the power of the numerator is greater the power of the denominator