Question

Test the series for convergence or divergence.

Answer 1

The ratio test is easier to carry out for this sum. You have

`\displaystyle\lim_(n\to\infty)\left|((3^(n+1)(n+1)!)/(11\cdot16\cdot21\cdot\cdots\cdot(5n+6)\cdot(5(n+1)+6)))/((3^nn!)/(11\cdot16\cdot21\cdot\cdots\cdot(5n+6)))\right| \\\\\\ = \lim_(n\to\infty)\left|((3(n+1))/((5(n+1)+6)))/((1)/(1))\right| \\\\\\ = \lim_(n\to\infty)\left|(3n+3)/(5n+11)\right| = \frac35 < 1`

which means the series converges (absolutely).