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Does the infinite geometric service diverge or converge?
1/5 + 1/10 + 1/20 + 1/40
A. it converges; it has a sum
B. It converges: it does not have a sum
To determine if the infinite geometric series converges or diverges, we need to check if the common ratio (ratio of consecutive terms) is between -1 and 1.
The common ratio for this series is calculated by dividing any term by its previous term:
1/10 ÷ 1/5 = 1/2
1/20 ÷ 1/10 = 1/2
1/40 ÷ 1/20 = 1/2
Since the common ratio is 1/2, which is between -1 and 1, the infinite geometric series converges.
To find the sum of the series, we use the formula S = a/(1-r), where a is the first term and r is the common ratio.
a = 1/5
r = 1/2
S = (1/5)/(1-1/2) = (1/5)/(1/2) = 2/5
Therefore, the infinite geometric series converges and has a sum of 2/5.
Answer: A. It converges; it has a sum.
To determine if the infinite geometric series converges or diverges, we can check if the common ratio, r, is between -1 and 1.
In this case, the common ratio can be found by dividing any term by its previous term.
Let's calculate the common ratio:
r = (1/10) / (1/5) = (1/10) * (5/1) = 5/10 = 1/2
Since the common ratio, 1/2, is between -1 and 1, the infinite geometric series converges.
Therefore, the correct answer is:
A. It converges; it has a sum.