Question

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

`sum_(n=0)^oo (n!)/999^n`

Answer 1

`sum_(n=0)^oo(n!)/(999^n)->` Does not converge

Explanation:

The ratio test tells us that if we have a series

`sum_(n=0)^ooa_n`

then if `lim_(n->oo)abs(a_(n+1)/a_n)<1`, the series converges.

Let's test `sum_(n=0)^oo(n!)/(999^n)`

`rArra_(n+1)/a_n=(((n+1)!)/(999^(n+1)))/((n!)/(999^n))=((n+1)!)/(999^(n+1))*(999^n)/(n!)=(n+1)/999`

`lim_(n->oo)abs((n+1)/999)=+oo`

Because this limit diverges, the ratio test tells us that the series does not converge.