Please help. TOTALLY confused. Convergence/divergence/finding sum of convergent series.

Question

Please help.  TOTALLY confused.  Convergence/divergence/finding sum of convergent series.

anonymous · Answer

$$\sum_{n=1}^{\infty}\frac{ 1 + 2^{n} }{ 3^n }$$

anonymous · Answer

Does this converge to 0 or to 2/3?  

I'm not sure if it converges to zero because the denominator goes to infinity faster than the numerator, or if, when I tried to do the test for divergence, my algebra was off.

I basically took a 3^n out of everything and ended up with 1/3^n + 2^n/3^n in the numerator and 1 in the denominator.  And I thought that the numerator would end up as 0 + 2/3.  So the result overall would be 2/3.  Is this correct?  Or is it really just zero?

anonymous · Answer

And I have no idea how to algebraically manipulate the problem to get it into Ar^n form in order to get the sum.  I would guess it would be the entire numerator of the original problem times the denominator raised to the negative first power.  But I don't know...