1. The series sums to 2/3.

1. The series sums to 6.

1. Factoring out a 3, this series is 3⋅(1 + 1/2 + 1/4 + 1/8 + …) The series in parentheses is a geometric series with ratio r = 1/2, converging because |r| < 1. This series sums to 1/(1 – r) = 1/(1 – (1/2)) = 2, so the original series sums to 3⋅2 = 6

1. The series sums to 1/6.

1. Factoring out a 1/9 from every term, we see the series is (1/9)⋅(1 + 1/3 + 1/9 + 1/27 + …) The series in parentheses is a geometric series with ratio r = 1/3, converging because |r| < 1. This series sums to 1/(1 – r) = 3/2, so the original series sums to (1/9)⋅(3/2) = 1/6

1. The series sums to 255

1. Recall for all r, 1 + r + r² + … + r^N-1= (1 – r^N)/(1 – r) Because 128= 2⁷, we see that 1 + 2 + 4 + … + 128 = (1 – 2⁸)/(1 – 2) = (1 – 256)/(1-2) = -255.

1. The series sums to 2ⁿ⁺¹– 1

1. Recall for all r, 1 + r + r² + … + r^N-1= (1 – r^N)/(1 – r) Then 1 + 2 + 4 + … + 2ⁿ= (1 – 2ⁿ⁺¹)/(1 – 2)= 2ⁿ⁺¹– 1.