Evaluate the geometric series or state that it diverges. one ninth 1 9plus+startfraction 7 over 81 endfraction 7 81plus+startfraction 49 over 729 endfraction 49 729plus+startfraction 343 over 6561 endfraction 343 6561plus + ...

The series converges, and its sum is 1/2.

If r > 1, the series is divergent. If r < 1, the series is convergent. In our sequence, r, the common ratio we multiply by to get the next term, is 7/9; therefore it is convergent.

To find the sum of a convergent series, we use the formula

a / (1-r), where a is the first term and r is the common ratio. We then have

1/9: (1-7/9) = 1/9:2/9 = 1/9*9/2 = 1/2