Question

Determine whether the geometric series converges. If it converges, find its sum. 27+9+3+1+............. Select the correct choice below and fill in any answer boxes within your choice.

A. If the series converges, provide the sum.

B. If the series diverges, state that it diverges.

C. If the series converges, provide the sum, and if it diverges, state that it diverges.

Answer 1

The geometric series 27 + 9 + 3 + 1 + ... converges with a common ratio of 1/3. Using the formula S = a / (1 - r), the series sum is found to be 40.5.

Step-by-step explanation:

We are asked to determine if the geometric series 27 + 9 + 3 + 1 + ... converges, and if it does, find its sum. A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). In this series, the common ratio is 1/3 because each number is a third of the previous number (for example, 9 is one third of 27).

The sum S of a convergent geometric series can be calculated using the formula S = a / (1 - r), where 'a' is the first term, and 'r' is the common ratio. Since the common ratio here is less than 1, the series converges. Substituting into the formula with a = 27 and r = 1/3:

S = 27 / (1 - 1/3) = 27 / (2/3) = 27 * (3/2) = 40.5. Therefore, the series converges, and the sum is 40.5.