Question

I need to determine the convergence/divergence of this infinite series:
\[\sum_{n=1}^{\infty} n! \div 2^{n}\]

Answer 1

That diverges. Wildly, as it turns out.

Answer 2

Yeah, my prof says it is diverging, but i need to know how it is derived.

Answer 3

You can use any number of tests. Ratio test would probably be easiest:
\[ \lim_{n -> \infty} \left| \frac{\frac{(n+1)!}{2^{n+1}}}{\frac{n!}{2^n}}\right| \]
\[= \lim_{n \rightarrow \infty} \left| \frac{n+1}{2} \right| \]
as n approaches infinity, that limit diverges, so the series diverges.

Answer 4

Wow, that's cleared some things! Thanks a lot!