Question

How can I prove this series alternating series converges(this is the answer)?

(-1)^2*(2/3)^n

I tried it this way: an = (2/3)^n, then i just broke it down. 2^n/(3^n) and i took the ratio of it and got 2/3 which does not equal to one which would mean the series diverges.. but that's obviously not how its done i guess
I assume you mean

(-1)^n * (2/3)^n

This is just a geometric series with r = -2/3

So, if you start with n=0, the sequence starts with 1, and

Sum = 1/(1-r) = 1/(1+2/3) = 3/5

In an alternating series, if the ratio |r| < 1 it converges.

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