Question

what is the sum for: inf. series from n=1 to inf. ((2^(n-1))/9^n)

Answer 1

ok,
writing the above in sigma notation, and doing a lot of simplifying, you can get:
=\[\sum_{n=1}^{\infty} (2/9)^n\]
now we're at home base. since the infinite geometric series sum equation is a/(1-r), where a is the first number and r is the ration between all the numbers,

we can easily plug 1 in the equation of the problem and get 1/9 as a
looking at the sigma notation, we see that the ration between the numbers is 2/9

after pluging in these numbers in,
we can find out that the sum is 1/7