Determine whether the geometric series converges or diverges.
27 + 18 + 12 + 8 + . . .
according to the given terms
a=27
r=2/3
S∞ = a/(1-r)
= 27/(1 - 2/3)
= 81
so what do you think?
Determine whether the geometric series 27 + 18 + 12 + 8 + ... converges or diverges, and identify the sum if it exists.
Well, this series seems to be getting smaller and smaller, so I guess you could say it's converging...just like my hairline! But in all seriousness, let's look at the common ratio of the series. By dividing each term by the previous term, we get 18/27 = 12/18 = 8/12 = 2/3. Since the absolute value of the common ratio, 2/3, is less than 1, we can conclude that this geometric series converges. So, it's slowly but surely getting closer and closer to a specific value.
To determine whether the geometric series converges or diverges, we need to check if the common ratio, denoted as "r", is between -1 and 1.
In the given series, the first term is 27, and each subsequent term is obtained by dividing the previous term by 1.5. Hence, the common ratio is 1.5.
Since the common ratio (1.5) is greater than 1, the series will diverge because it will continue to get larger and larger with each term.
To determine whether the geometric series converges or diverges, we need to look at the common ratio.
The common ratio (r) is found by dividing any term in the series by its preceding term. In this case, we can see that:
18 ÷ 27 = 2/3
12 ÷ 18 = 2/3
8 ÷ 12 = 2/3
...
Since the common ratio (r) is constant and equal to 2/3, we can determine whether the series converges or diverges by looking at the value of r.
For a geometric series to converge, the absolute value of r must be less than 1. In this case, |2/3| = 2/3, which is less than 1. Therefore, the geometric series converges.
To find the sum of the converging geometric series, we can use the formula:
S = a / (1 - r)
where:
S = sum of the series
a = first term of the series
r = common ratio
In this case, the first term (a) is 27 and the common ratio (r) is 2/3. Plugging these values into the formula, we get:
S = 27 / (1 - 2/3)
Simplifying, we have:
S = 27 / (3/3 - 2/3)
S = 27 / (1/3)
To divide by a fraction, we multiply by its reciprocal, so:
S = 27 * (3/1)
S = 81
Therefore, the sum of the given converging geometric series is 81.