Determine whether the series is convergent or divergent If i

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

1/3 + 2/9 + 1/27 + 2/81 +1/243 + 2/729 + . . .

Solution

It is convergent. Consider this as the sum of the sum of two geometric sequences:
1/3 + 1/27 + 1/243 ...
2/9 + 2/81 + 2/729 ...
With each series, we have the same ratio, 1/9
|1/9| < 1 implies convergence
We use formula, a/1-r, where a is the first term and r is the ratio.
1/3/(1 - -1/9) = 1/3/(8/9) = 1/3(9/8) = 3/8
2/9/(1 - -1/9) = 2/9/(8/9) = 2/9(9/8)=1/4

3/8 + 1/4 = 5/8