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

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

1 Answer
Mar 31, 2018

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.