Use the limit comparison test to prove convergence or divergence for the series sum from n=1 to infinity for ((5n^3)+1)/((2^n)((n^3)+n+1))
The limit comparison test says that if you have two positive series, sum An and sum Bn, let C=lim n to infinity of An/Bn and (i) if 0< C < infinity; then both series converge or both diverge; (ii) if C = 0 and sum Bn converges, so does sum An; (iii) if C = infinity and sum Bn diverges, so does sum An.
The Attempt at a Solution
For An, I used L'Hospital's rule (twice) to get lim as n to infinity of 30/infinity, which equals 0. For Bn, I chose Bn = 2^n, which is a divergent geometric series with an r of 2 (>1). This didn't help me apply the limit comparison test, so I changed my Bn to the convergent geometric series 0.5^n, which has an r of 0.5 (<1). This also didn't help me apply the limit comparison test. Can someone help me please? Thanks so much in advance.