Diverges or converges?

Question

(infinity symbol)

∑ (n^2)/(5n^2+4)

n=1

 

How do you know if it diverges. Please explain step by step. 

Answer 1

The reason is that the nth term does not approach 0 as n -> ∞. Recall the nth term test: If a series ∑ a_n converges, then lim a_n=0. With a_n= n²/(5n²+4), we have lim a_n = 1/5 ≠ 0. So by the nth term test, your series diverges.