Does this series converge ?

Question

Zero to infinity∑ (-1)ⁿ * ((5n²+7)/(7n²+10))^ⁿ 

 

I think I have to use the root test but not sure. What test do I use? If it is the root test, does the square root cancel the both n's on each bracket ?

Answer 1

Root test will work.

Since a_n = (-1)ⁿ[(5n²+7)/(7n²+10)]ⁿ, the n^th-root of |an| is (5n²+7)/(7n²+10) which has a limit of 5/7 < 1 as n →∞.

The series converges.