Question

Use Ratio Test to find the convergence of the following series?

`sum_(n=1)^oo ((n+3)!)/ ((3!)(n!)(3^n)`

Answer 1

The series:

`sum_(n=0)^oo (( n+3)!) / ( ( 3!)(n!)(3^n))`

is convergent.

Explanation:

Evaluate the ratio:

`a_(n+1)/a_n = abs ( ( ( (n+1+3)!)/((3!)((n+1)!)3^(n+1)) ) / ( (( n+3)!) / ( ( 3!)(n!)(3^n)))`

`a_(n+1)/a_n = ((n+4)!)/((n+3)!) (3!)/(3!) (n!)/((n+1)!) 3^n/3^(n+1)`

`a_(n+1)/a_n =1/3 (n+4)/(n+1)`

Hence:

`lim_(n->oo) a_(n+1)/a_n = 1/3`

and the series is convergent.