## anonymous 5 years ago Determine whether the sequence converges or diverges. If it converges, nd the limit n!/2^n

1. anonymous

You can use the ratio test. $\lim_{n->\infty}\left|\frac{a_{n+1}}{a_n} \right|=\lim_{n->\infty}\frac{(n+1)!/2^{n+1}}{n!/2^{n}}=\lim_{n->\infty}\frac{(n+1)n!}{2 \times 2^n}\frac{2^n}{n!}$$=\lim_{n->\infty}\frac{n}{2} \rightarrow \infty$ The sequence does not converge since the ratio in the limit is not less than 1.

2. anonymous

nice!

3. anonymous

Thanks ;)

4. anonymous

$\frac{n+1}{2}$

5. anonymous

A question.. So does it mean that you cant test some values as N... and see whats happening?

6. anonymous

Can you rephrase that? I'm not sure what you mean.

7. anonymous

n!/2^n... Put in some values, as a check.. ..n= 1, 2, 3

8. anonymous

And see if it diverges or converges

9. anonymous

I hope you didnt misunderstand me, it was just a question, if i would in some values, in the funktion , n!/2^n .. could i see right away if it diverges or converges..

10. anonymous

Ah I see. No, because that's not actually testing for ALL n. The tests are derived from the definition of convergence.

11. anonymous

You could get an idea, but it wouldn't prove it. That's all.

12. anonymous

Okey, so u have to, show it for all n. like u did

13. anonymous

Exactly.

14. anonymous

Well, thx.

15. anonymous

No worries.

16. anonymous

Hehe, I thought this was XiaoHong's question.

17. anonymous

So did i ;)

18. anonymous

thanks

19. anonymous

Welcome.