Final answer:
The sum of the geometric series 1 - 3 + 3^2 - 3^3 + ... - 3^-29 can be found using the formula for the sum of a geometric series with the common ratio -3 and the number of terms being 30.
Step-by-step explanation:
The sum of the geometric series 1 - 3 + 32 - 33 + ... - 3-29 involves alternating signs with each term being the cube of the previous one (except the first term which is just 1). This pattern can be represented as a geometric series with a common ratio of -3. To find the sum of a geometric series, we can use the formula:
S = a1 * (1 - rn) / (1 - r)
where a1 is the first term of the series, r is the common ratio, and n is the number of terms in the series. For this series, a1 = 1, r = -3, and n = 30 since the power of the last term is -29. Plugging these values into the formula gives the sum of the series. Remember that due to the nature of this series, when computing rn, we must cubing of exponentials, which in this case just means taking -3 to the power of -29.