Convergence of Infinite Series

[americanforest]

Question: Test for convergence:

$$\sum\frac{n!}{10^n}$$

(the sum is from 1 to infinity)

I tried using

$$\frac{n^n}{10^n}\geq\frac{n!}{10^n}\geq\frac{n}{10^n}$$

and showing that either the first one was convergent or the last one was divergent using various tests but didn't get anywhere.

Any hints?

 

[J0EBL0W]

Try using the Ratio Test. That's what I first try to do anytime I see n!

 

[americanforest]

Ratio Test:

$$\frac{(n+1)!}{10^{n+1}}\frac{10^n}{n!}=\frac{n+1}{10}$$

Since that's more than 1 as n goes to infinity it diverges. Am I right?

 

[americanforest]

I did a bunch of these. I need to get a good grade so I will just post them up here with my answers and if I got one wrong please just let me know to look over it again.

1. $$\sum\frac{1}{ln(n)}$$ (sum from 2 to inf.) diverges

b/c 1/ln(n)>1/n which diverges

2. $$\sum\frac{1}{2n(2n+1)}$$ converges

b/c 1/n^2 converges

3. $$\sum\frac{1}{(n(n+1))^{.5}}$$ diverges

b/c 1/n diverges and then limit comparison test to show this diverges too.

4. $$\sum\frac{1}{2n+1}$$ diverges

b/c 1/2n diverges and then limit comparison to show this diverges too.

I'd really appreciate it if somebody could check me on these. Thanks.

 

[Office_Shredder]

Those all look right to me