Thread: Checking this series for convergence

1. Checking this series for convergence

Hi all,

I'm new to this, and am looking at the following series:

$\sum\limits_{n = 1}^\infty{\frac{{1 \cdot 3 \cdot 5(2n - 1)}}{{1 \cdot 4 \cdot 7(3n - 2)}}}$

I decided to apply the ratio test and came up with the following:

$\rho= \mathop {\lim }\limits_{n \to \infty } \frac{{1 \cdot 3 \cdot 5(2(n + 1) - 1)}}{{1 \cdot 4 \cdot 7(3(n + 1) - 2)}} \cdot \frac{{1 \cdot 4 \cdot 7(3n - 2)}}{{1 \cdot 3 \cdot 5(2n - 1)}} = \mathop {\lim }\limits_{n \to \infty } \frac{{(3n - 2)(2n + 1)}}{{(2n - 1)(3n + 1)}}$

This gives me a rho value of 1, telling me absolutely nothing. My question is if my approach was valid, or if I should have cancelled some terms that I didn't cancel. Did I do this right?

Also, how would you approach this series to test convergence?

2. Thanks Mr. Fantastic...what happened with the latex? I was checking to see where I made a mistake.... What is the correct type of latex for here? I should have had that formatted for LaTeX 2.09 and greater.

Edit: Nevermind....I see the other posts on the board regarding this issue now.

3. Originally Posted by Malaclypse
Hi all,

I'm new to this, and am looking at the following series:

$\sum\limits_{n = 1}^\infty{\frac{{1 \cdot 3 \cdot 5(2n - 1)}}{{1 \cdot 4 \cdot 7(3n - 2)}}}$

I decided to apply the ratio test and came up with the following:

$\rho= \mathop {\lim }\limits_{n \to \infty } \frac{{1 \cdot 3 \cdot 5(2(n + 1) - 1)}}{{1 \cdot 4 \cdot 7(3(n + 1) - 2)}} \cdot \frac{{1 \cdot 4 \cdot 7(3n - 2)}}{{1 \cdot 3 \cdot 5(2n - 1)}} = \mathop {\lim }\limits_{n \to \infty } \frac{{(3n - 2)(2n + 1)}}{{(2n - 1)(3n + 1)}}$

This gives me a rho value of 1, telling me absolutely nothing. My question is if my approach was valid, or if I should have cancelled some terms that I didn't cancel. Did I do this right?

Also, how would you approach this series to test convergence?
Your cancelling is wrong. You should have got $\frac{2n+1}{3n+1}$.

4. Thank you. That's where I figured I was going horribly wrong, but where I'm struggling is understanding just how those terms cancel. Is it basically because the (3n-2) and the (2n-1) exist in previous terms?

I'm having a hard time seeing it as an obvious fact....but I suppose that's what practice is for.

5. Originally Posted by Malaclypse
Thanks Mr. Fantastic...what happened with the latex? I was checking to see where I made a mistake.... What is the correct type of latex for here? I should have had that formatted for LaTeX 2.09 and greater.

Edit: Nevermind....I see the other posts on the board regarding this issue now.