# Thread: convergence/divergence(urgent help needed plz!)

1. ## convergence/divergence(urgent help needed plz!)

Use any method you can to decide the convergence of
i)

ii)

iii)

iv)

2. Hi
Originally Posted by matty888
Use any method you can to decide the convergence of
i)
Evaluate $\lim_{n\to \infty}\frac{\sqrt{n}}{\ln n}$

ii)
Evaluate $\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$

iii)
It's a geometric series

iv)
Use the ratio test. (it's often useful with fractions involving factorials)

Hello could you do the first one for me please im really stuck!thanks

4. Originally Posted by matty888
Hello could you do the first one for me please im really stuck!thanks
Do you realise that $\sum a_n$ diverges if $\lim_{n \rightarrow \infty} a_n \neq 0$ ?

5. Originally Posted by mr fantastic
Do you realise that $\sum a_n$ diverges if $\lim_{n \rightarrow \infty} a_n \neq 0$ ?
$\lim_{n\rightarrow{\infty}}\frac{\sqrt{n}}{\ln{n}}$ is obviously infinity, but how to prove it?

6. Originally Posted by disclaimer
$\lim_{n\rightarrow{\infty}}\frac{\sqrt{n}}{\ln{n}}$ is obviously infinity, but how to prove it?
Well L'Hopitals rule will work

RonL

7. Originally Posted by CaptainBlack
Well L'Hopitals rule will work

RonL
I've never seen L'Hospital's rule used to calculate a limit of a sequence (sequences don't have derivatives).

8. Originally Posted by disclaimer
I've never seen L'Hospitals rule used to calculate a limit of a sequence (sequences don't have derivatives).

What do you mean? Does it matter whether it is a limit of a sequence or limit of any other function

9. Originally Posted by disclaimer
I've never seen L'Hospital's rule used to calculate a limit of a sequence (sequences don't have derivatives).
Technically you are looking at the limit of $\frac{\sqrt{x}}{\ln(x)}$ but the two limits are equal (as long as you go from the continuous to the discrete anyway). (If you are not convinced suppose otherwise and you will rapidly find a contradiction)

RonL

10. Originally Posted by CaptainBlack
Technically you are looking at the limit of $\frac{\sqrt{x}}{\ln(x)}$ but the two limits are equal (as long as you go from the continuous to the discrete anyway). (If you are not convinced suppose otherwise and you will rapidly find a contradiction)

RonL
When I was doing series, to be able to use L'Hopital's rule I would simply say:

$a_n$= $f(n)$

I found that my teacher accepted that, then I could use L'H because I was dealing with a continuous function.

11. Originally Posted by flyingsquirrel
Hi

Evaluate $\lim_{n\to \infty}\frac{\sqrt{n}}{\ln n}$
Evaluate $\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$

It's a geometric series

Use the ratio test. (it's often useful with fractions involving factorials)
This one $\lim_{n \to {\infty}}\bigg(1+\frac{1}{n}\bigg)^{n}$ is the famous definiton for $e\approx{2.71828183...}$..which means the sum diverges by the n-th term test...if you want to see how this was derived(the limit) look here http://www.mathhelpforum.com/math-he...-tutorial.html

12. Originally Posted by Mathstud28
This one $\lim_{n \to {\infty}}\bigg(1+\frac{1}{n}\bigg)^{n}$ is the famous definiton for $e\approx{2.71828183...$..which means the sum diverges by the n-th term test...if you want to see how this was derived(the limit) look here http://www.mathhelpforum.com/math-he...-tutorial.html
Hello,

I sincerely think that Flyingsquirrel knows how to do
He didn't say he couldn't do it, he just gave the way to do ^^

Are you paying the advertisings ?

13. Originally Posted by Moo
Hello,

I sincerely think that Flyingsquirrel knows how to do
He didn't say he couldn't do it, he just gave the way to do ^^

Are you paying the advertisings ?
Oh...I know Flyingsqurriel can do it...I meant to quote the poster...