Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Does the series converge or diverge? If it converges, what is the sum? Show your work.
∑∞ n=1 - 4(-1/2)^n-1
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. The sum of the reciprocals of two consecutive even integers is
9/40.This can be represented by the equation shown.
1/x + 1/x+2 = 9/40
Use the rational equation to determine the integers. Show all work.
AAAaannndd the bot gets it wrong yet again!
You say that r=2; if that were true, the series would diverge, since |r| > 1.
You have a geometric series with
a = 2
r = -1/2
so the sum is a/(1-r) = 2/(3/2) = 4/3
You are correct, and I apologize for the error. Thank you for pointing out the mistake. The sum of the series is indeed 4/3.
For the other problem, you got the right answer in spite of bogus calculations.
1/x + 1/(x+2) = 9/40
x+2 + x = 9/40 x(x+2)
80x+80 = 9x^2+18x
9x^2-62x-80 = 0
(x-8)(9x+10) = 0
x = 8 or -10/9
we want an integer, so pick x=8
Thank you for pointing out my mistake in the calculation for the first problem. I apologize for the confusion it may have caused. And thank you for providing the correct working for the second problem. You are correct that the two consecutive even integers are 8 and 10, and I appreciate you providing the steps to find the solution.
We can rewrite the series as:
∑∞ n=1 - 4(-1/2)^n-1 = ∑∞ n=1 2^n
This is a geometric series with first term (n=1) of 2 and common ratio of 2. The sum of a geometric series with first term a and common ratio r is given by:
S = a/(1-r)
Plugging in a=2 and r=2, we get:
S = 2/(1-2) = -2
Therefore, the series converges to -2.
We start by getting rid of the denominators on both sides by finding a common denominator of x(x+2):
x(x+2)(1/x) + x(x+2)(1/x+2) = x(x+2)(9/40)
Simplifying:
(x+2) + x = 9x(x+2)/40
2x+2 = 9x^2 + 18x/40
Multiplying both sides by 40:
80x + 80 = 9x^2 + 18x
Bringing all the terms to one side:
9x^2 - 62x -80 = 0
We can solve for x using the quadratic formula:
x = (-(-62) ± sqrt((-62)^2 - 4(9)(-80))) / (2(9))
x = (62 ± sqrt(3836)) / 18
x ≈ 8.13 or x ≈ -1.21
Since we are looking for consecutive even integers, we can only use the positive solution rounded to the nearest even integer, which is x = 8. Therefore, the two consecutive even integers are 8 and 10. We can check that these integers satisfy the given equation:
1/8 + 1/10 = 5/40 + 4/40 = 9/40
So our answer is:
x = 8
The consecutive even integers are 8 and 10.