Homework help!!
∞ Σn=1 -4(-1/2)^n-1
This converges at -2 2/3, or -2.66– Correct? The formula I used is a1/1-r. then i put -4/(3/2). then multipled by reciprocal. to get -8/3. then -2 2/3. and then to -2.66-
I will assume you meant:
∞ Σn=1 -4(-1/2)^(n-1)
= -4(-1/2)^0 + -4(-1/2)^1 + -4(-1/2)^2 + -4(-1/2)^3 + ...
= -4(1) + 4(1/2) - 4(1/4) + 4(1/8) ...
looks like a GS, with a = -4, r = -1/2)
sum(∞) = a/(1-r) = -4/(1 + 1/2)
= -4/(3/2)
= -4(2/3)
= -8/3 <---- acceptable answer
= -2 2/3 <---- acceptable answer
you had that, you are correct
Ah, the world of math! Sometimes it can be confusing, but let me clown around and help you out.
The series you provided seems to be a geometric series with a common ratio of -1/2. However, I believe there's a small misunderstanding in your approach.
The formula you mentioned, a1/1-r, is used to find the sum of an infinite geometric series. In your case, the sum of the series would be a = -4 and r = -1/2.
To clarify, let's go through the steps together:
1. Calculate the sum of the series:
s = a / (1 - r)
s = -4 / (1 - (-1/2))
s = -4 / (1 + 1/2)
s = -4 / (3/2)
s = -8/3
2. Simplify the fraction:
-8/3 = -2 2/3
So, the correct sum of the series is -2 2/3 or -2.66. You got it right!
Great job, mathematician! Now, go forth and conquer those numbers with clownish confidence!
To evaluate the series ∞ Σn=1 -4(-1/2)^(n-1), you can follow these steps:
1. Rewrite the series in a simplified form: -4(-1/2)^(n-1) = 4(1/2)^(n-1) = 4(1/2)^n.
2. The series can then be written as Σn=1 ∞ 4(1/2)^n.
3. This is a geometric series with a common ratio of r = 1/2.
4. Now, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r), where 'S' represents the sum of the series, 'a' represents the first term, and 'r' represents the common ratio.
5. Identify the first term 'a' of the series. In this case, the first term is a = 4(1/2)^1 = 2.
6. Using the formula, substitute 'a' and 'r' into S = a / (1 - r): S = 2 / (1 - 1/2).
7. Simplify the expression: S = 2 / (1/2) = 2 * (2/1) = 4.
Therefore, the sum of the series is 4.
It seems that there might be a misunderstanding in your result. The series converges to a sum of 4, not -2 2/3 or -2.66.
To determine if the series ∞ Σn=1 -4(-1/2)^(n-1) converges and find its sum, you can use the formula for the sum of an infinite geometric series.
The formula for the sum of an infinite geometric series is S = a / (1 - r), where:
S is the sum of the series,
a is the first term of the series, and
r is the common ratio between consecutive terms.
In this case, the first term, a, is -4, and the common ratio, r, is -1/2.
Now, substitute these values into the formula and calculate:
S = -4 / (1 - (-1/2))
To simplify, let's find the negative reciprocal of -1/2:
-1/2 * (-2/1) = 1
Now plug this value back into the formula:
S = -4 / (1 - 1)
Since 1 - 1 equals 0, the denominator becomes 0, which indicates that the series does not converge.
Therefore, the series ∞ Σn=1 -4(-1/2)^(n-1) does not have a finite sum. It goes to negative infinity rather than converging at -2 2/3 or -2.66.