Can you determine the convergence of this sum notation series for math homework?

[chemnoob.]

Homework Statement

Determine whether series diverges of converges (conditionally or absolutely)
(1+2)/(1+3) + (1+2+4)/(1+3+9) + (1+2+4+8)/(1+3+9+27) ...

Homework Equations

The Attempt at a Solution

I'm assuming that I would have to put it into sum notation but I am struggling with that.

 

[Mark44]

Homework Statement

Determine whether series diverges of converges (conditionally or absolutely)
(1+2)/(1+3) + (1+2+4)/(1+3+9) + (1+2+4+8)/(1+3+9+27) ...

Homework Equations

The Attempt at a Solution

I'm assuming that I would have to put it into sum notation but I am struggling with that.

It looks to me like the general term of your series is
$$a_n = \frac{\sum_{i = 0}^n 2^i}{\sum_{i = 0}^n 3^i}$$

Can you work with that?

 

[chemnoob.]

What test should I try using?

 

[Mark44]

Both series are geometric series, so you should be able to get a simpler expression for a_n. Once you do that, you could take the limit of a_n as n goes to infinity. If you get a nonzero value, you know the series diverges. If you get zero, then you need to use more tests, such as the ratio test, limit comparison test, etc.

 

[chemnoob.]

It looks to me like the general term of your series is
$$a_n = \frac{\sum_{i = 0}^n 2^i}{\sum_{i = 0}^n 3^i}$$

Can you work with that?

I think you are missing a sum at the beginning of all of that .. but I still don't know how you would go about solving that!

 

[Mark44]

What I wrote is the general term of the series, not the whole series. See if you can write a_n without the two summations, using what you know about finite geometric series.