A series involving factorial

Question

Series from $n=0$ to infinity of $n!/1000^n$

I know the limit of $n!$ is infinity and $1000^n$ is also infinity. In this regard, I really don't see how L'Hopital's rule can work in this case. How do I tackle this problem ?

Answer 1

Since$$\lim_{n\to\infty}\frac{\frac{(n+1)!}{1000^{n+1}}}{\frac{n!}{1000^n}}=\lim_{n\to\infty}\frac n{1000}=\infty,$$your series diverges, by the ratio test.

Answer 2

For $n >2000$ we have $\frac {n!} {(1000)^{n}} >\frac {(2000)(2001)...n} {(1000)^{n}} >2^{n-2000}$ so the general term of the series tends to $\infty$.

Answer 3

When you go from the term $n$ to the term $n+1$, you multiply by $n+1$ and divide by $1000$. So the numerator will eventually "win" and make the terms larger and larger.

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More rigorously, for $n>1000$,

$$t_n:=\frac{n!}{1000^n}\ge t_{1000}1.001^{n-1000},$$ forming a diverging geometric series.