Infinite Series Homework: Determine Convergence/Divergence

[mreaume]

Homework Statement

Determine whether the series diverges or converges.

(1+2) / (1+3)+ ((1+2+4)/(1+3+9))+ ((1+2+4+8)/(1+3+9+27)) + ...

The Attempt at a Solution

I have split up the series into two (denominator and numerator):

an = (1+2) + (1+2+4) + (1+2+4+8)+... = (1)n + 2n + 4(n-1) + ...
bn = (1+3) + (1+3+9)+... = (1)n + (3)n + (9)(n-1)+... = (1)n + 3n + 9(n-1) + ...I don't know how to keep going. I suspect that the ratio test will come in handy later but am not sure how to apply it with the given series above. Any help would be appreciated. Thanks.

 

[Office_Shredder]

How is evaluating a_n and b_n supposed to help? a_n/b_n is not the same as the partial sum of the series you have been given.

You might want to think about a short way to write down
1+2+4+8+16+...
for any finite term.

 

[mreaume]

Well we can write 1+2+4+8+... as

the sum from n=0 to n=inf of 2^n.

And similarly we can write 1+3+9+...

as the sum from n=0 to n=inf of 3^n.

So can we say that the series is (2/3)^n? From n=0 to n=inf?