posted by Isaac on .
How can I prove this series alternating series converges(this is the answer)?
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.