# Thread: can someone verify if i'm right about these series

1. ## can someone verify if i'm right about these series

I have to find whether these series converge.

$\displaystyle \sum {\frac {n+1} {n^2 + 1}}$ by using the limit comparison test, and using $\displaystyle \sum {b_n}$ = $\displaystyle \frac {1} {n}$ i found that it diverges.

As for this example: $\displaystyle \sum {\frac {n!} {n^n}}$ i used the ratio test and found it diverges also.

Can someone check if i am correct, and also help me with this one
$\displaystyle \sum {\frac {1} {2^n - 1 + cos^2(n^3)}}$

2. Originally Posted by Cato
I have to find whether these series converge.

$\displaystyle \sum {\frac {n+1} {n^2 + 1}}$ by using the limit comparison test, and using $\displaystyle \sum {b_n}$ = $\displaystyle \frac {1} {n}$ i found that it diverges.
This diverges so you are correct.

As for this example: $\displaystyle \sum {\frac {n!} {n^n}}$ i used the ratio test and found it diverges also.
This is wrong. The ratio that you should get is $\displaystyle \left( \frac{n}{n+1} \right)^n$. The limit of this is $\displaystyle 1/e < 1$. Therefore it converges.

Can someone check if i am correct, and also help me with this one
$\displaystyle \sum {\frac {1} {2^n - 1 + cos^2(n^3)}}$
$\displaystyle \frac{1}{2^n - 1 + \cos^2(n^3)} \leq \frac{1}{2^n - 1}$

3. Thank you, so $\displaystyle \frac{1}{2^n - 1}$ would then diverge when n tends to infinity?

4. Originally Posted by Cato
I have to find whether these series converge.

$\displaystyle \sum {\frac {n+1} {n^2 + 1}}$ by using the limit comparison test, and using $\displaystyle \sum {b_n}$ = $\displaystyle \frac {1} {n}$ i found that it diverges.

As for this example: $\displaystyle \sum {\frac {n!} {n^n}}$ i used the ratio test and found it diverges also.

Can someone check if i am correct, and also help me with this one
$\displaystyle \sum {\frac {1} {2^n - 1 + cos^2(n^3)}}$
You got it right for the first.
For the last one, use the fact that $\displaystyle \cos x\leq 1$. If i give you a little bit more hint it will be obvious. So try first and let see if you get it or not

Edit: sorry I don't know if someone already give a reply (the ad always deceives me as if there is no reply to the post).
I think it is also bound by 1/2^n which is converges (geometric series).