1. anonymous

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

2. anonymous

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?

3. anonymous

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...