Question

Converges or Diverges?

Answer 1

Must use the ratio or root test:

n^n/n!

And

1.19^n/n^5

Answer 2

Wait, actually for the second one can use comparison or lim as n approaches infinity as well.

Answer 3

both of them diverges

Answer 4

for first you use Stirling approximation ...
for second, you just use L'hopital rule .... exponential functions always wins over polynomial functions.

Answer 5

\[\frac{n^n}{n!}=\frac{n\cdot n\cdots n}{n(n-1)\cdots1}=\frac{n}{n}\frac{n}{n-1}\cdots\frac{n}{1}>n\to \infty \text{ as }n\to\infty\]