Calculus of a Single Variable, Chapter 9, 9.2, Section 9.2, Problem 14

sum_(n=1)^oo (n!)/2^n
To verify if the series diverges, apply the ratio test. The formula for the ratio test is:
L = lim_(n->oo) |a_(n+1)/a_n|
If L<1, the series converges.
If L>1, the series diverges.
And if L=1, the test is inconclusive.
Applying the formula above, the value of L will be:
L = lim_(n->oo) |(((n+1)!)/2^(n+1))/ ((n!)/2^n)|
L= lim_(n->oo) |((n+1)!)/2^(n+1) * 2^n/(n!)|
L=lim_(n->oo) | ((n+1)*n!)/(2*2^n) * 2^n/(n!)|
L = lim_(n->oo) | (n+1)/2|
L = 1/2 lim_(n->oo) |n+ 1|
L=1/2 * oo
L=oo
Therefore, the series diverges.