Does the following series converges or diverges?

Question

∞

∑ (n²-3n+5)/(n³+4n-1)

n=1

Answer 1

∞
∑ (n²-3n+5)/(n³+4n-1)
n=1

As n gets larger and larger, the n² in the numerator dominates all other terms there, and the n³ in the denominator dominates all other terms there.

 

The series

∞
∑ (n2-3n+5)/(n3+4n-1) →  ∑ n²/ n³ = ∑1/ n  which is the familiar divergent p series with p =1.
n

 

The series therefore diverges.