Estimate the sum of the alternating series the sum from n equals 0 to infinity of the quotient of negative 1 raised to the nth power and the quantity 2 times n factorial within an error of 0.0001.
Type your answer in the space below and give six decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482317). (10 points)
well, cos(x) = 1 - x^2/2! + x^4/4! - ...
Your series is
1 - 1/2! + 1/4! - ...
so, what do you think?
To estimate the sum of the alternating series, we can use the Alternating Series Estimation Theorem. The theorem states that if we have an alternating series of the form (-1)^n * a_n, where a_n is a positive sequence that decreases monotonically to zero as n approaches infinity, then the error in the estimation of the sum after n terms can be upper-bounded by the next term in the series.
In this problem, the alternating series is given by (-1)^n * (1 / (2n!)). To estimate the sum within an error of 0.0001, we need to find the minimum value of n such that the absolute value of the next term in the series, |(-1)^(n+1) * (1 / (2(n+1)!))|, is less than 0.0001.
Let's calculate the terms of the series until we find the first term that satisfies this condition.
n = 0: (-1)^0 * (1 / (2*0!)) = 1 / 1 = 1
n = 1: (-1)^1 * (1 / (2*1!)) = -1 / 2 = -0.5
n = 2: (-1)^2 * (1 / (2*2!)) = 1 / 4 = 0.25
n = 3: (-1)^3 * (1 / (2*3!)) = -1 / 12 = -0.083333
n = 4: (-1)^4 * (1 / (2*4!)) = 1 / 48 = 0.020833
n = 5: (-1)^5 * (1 / (2*5!)) = -1 / 240 = -0.004166...
n = 6: (-1)^6 * (1 / (2*6!)) = 1 / 1440 = 0.000694...
It seems that by n = 6, the value of the term is less than 0.0001. Therefore, the sum after 6 terms should be a good estimate within the desired error range.
Now let's calculate the partial sum of the series up to n = 6.
S_6 = (-1)^0 * (1 / (2*0!)) + (-1)^1 * (1 / (2*1!)) + (-1)^2 * (1 / (2*2!)) + (-1)^3 * (1 / (2*3!)) + (-1)^4 * (1 / (2*4!)) + (-1)^5 * (1 / (2*5!)) + (-1)^6 * (1 / (2*6!))
S_6 = 1 - 0.5 + 0.25 - 0.083333 + 0.020833 - 0.004166 + 0.000694
S_6 ≈ 0.678087
Therefore, the estimated sum of the alternating series, rounded to six decimal places, is 0.678087.