Consider the infinite geometric series below.
a. Write the first 4 terms of the series
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.
∞
∑ (-2)^n-1
n=2
not sure how to do this at all
∞
∑ (-2)^n-1
n=2
= -2 + 4 - 8 + 16 - 32 + 64 - ...
= (-2 + 4) + (- 8 + 16) + ( - 32 + 64) + ( - ...
= 2 + 8 + 32 + .....
So, what do you think?
i have no clue
No problem! I can help you step-by-step with this problem.
An infinite geometric series is defined as a series in which each term is found by multiplying the previous term by a constant ratio. In this case, the series is given as:
∞
∑ (-2)^(n-1)
n=2
a. To find the first 4 terms of the series, we can substitute the values of n from 2 to 5:
n = 2: (-2)^(2-1) = (-2)^1 = -2
n = 3: (-2)^(3-1) = (-2)^2 = 4
n = 4: (-2)^(4-1) = (-2)^3 = -8
n = 5: (-2)^(5-1) = (-2)^4 = 16
The first four terms of the series are: -2, 4, -8, 16.
b. To determine if the series diverges or converges, we need to check the absolute value of the common ratio, which is -2 in this case. If the absolute value of the common ratio is less than 1, then the series converges. If the absolute value is greater than or equal to 1, then the series diverges.
In this case, since the absolute value of -2 is equal to 2, which is greater than 1, the series diverges.
c. As the series diverges, there is no finite sum.
So, the answers to the questions are:
a. The first 4 terms of the series are -2, 4, -8, 16.
b. The series diverges.
c. The series does not have a finite sum.
I hope this helps! Let me know if you have any further questions.
To find the first 4 terms of the given infinite geometric series, we need to substitute the values of n into the formula and calculate the corresponding terms.
The given series is ∑ (-2)^(n - 1), where n starts from 2 and goes to infinity. Let's find the first four terms:
When n = 2:
Term 1: (-2)^(2 - 1) = (-2)^1 = -2
When n = 3:
Term 2: (-2)^(3 - 1) = (-2)^2 = 4
When n = 4:
Term 3: (-2)^(4 - 1) = (-2)^3 = -8
When n = 5:
Term 4: (-2)^(5 - 1) = (-2)^4 = 16
a. The first four terms of the series are: -2, 4, -8, 16.
To determine whether the series converges or diverges, we need to check the common ratio, |r|, where r is the ratio between consecutive terms.
In this case, the common ratio, r, is -2. Since the absolute value of -2 is | -2 | = 2, which is not less than 1, the series diverges. When the absolute value of the common ratio is greater than or equal to 1, the series does not have a sum and diverges.
b. The given series diverges.
To find the sum of an infinite geometric series, we can use the formula:
Sum = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio.
In this case, the first term, 'a', is -2, and the common ratio, 'r', is -2.
c. Since the series diverges and does not have a sum, we cannot find the sum for this series.