a_(n)=(5n^(2)+14n)/(3n^(4)-5n^(2)-18),b_(n)=(5)/(3n^(2))
Calculate the limit.
(Give an exact answer. Use symbolic notation and fractions where needed. Enter DNE if the limit does not exist.)
\lim_(n->\infty )(a_(n))/(b_(n))=
Determine the convergence or divergence of \sum_(n=1)^(\infty ) a_(n).
\sum_(n=1)^(\infty ) a_(n) converges by the Limit Comparison Test because \lim_(n->\infty )(a_(n))/(b_(n)) is finite and \sum_(n=1)^(\infty ) b_(n) converges.
\sum_(n=1)^({(:[n=1]),(\infty ):}) a_(n) diverges by the Limit Comparison Test because \lim_(n->\infty )(a_(n))/(b_(n)) is finite and \sum_(n=1)^({(:n=1):}) b_(n) diverges.
It is not possible to use the Limit Comparison Test to determine the convergence or divergence of \sum_(n=1)^(\infty ) a_(n).
\sum_(n=1)^(\infty ) a_(n) converges by the Limit Comparison Test because \lim_(n->\infty )(a_(n))/(b_(n)) is finite and \sum_(n=1)^(\infty ) b_(n) diverges.
