series sum?

[hadimotamedi]

Dear All
Can you please let me know how can I calculate the summation of the following series when n approaches infinity?
Sn=1/2 + 1/3 + 2/9 + ... + an
where an = (2^(n-2))/(3^(n-1))
Thank you

[Ell]

Starting from the second term, this is a geometric series with ratio = 2/3. Hence the summation equals the first term divided by 1 - the ratio.
So 1/3 + 2/9 + ... = (1/3)/(1-(2/3)) = 1
The entire series 1/2 + 1/3 + 2/9 + ... = 1/2 + 1 = 3/2

[hadimotamedi]

Thank you very much for your help. I got the point.

[vonLipwig]

You don't really need to treat the first term separately if you don't want to - the whole thing is a geometric series. That is, rather than multiplying 1/3 by 3 and adding 1/2, giving 3/2, you could just multiply the first term, 1/2, by 3, to get 3/2.

[Ell]

Yes, you are right, the whole series is geometric. With the first term equal to 1/2, the whole sum is (1/2)/(1/3) = 3/2.