Check my answer! How many terms of the series the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and n do we need to add in order to find the sum with an absolute value of its error to be less than 0.05? (4 points) Is the answer 20?
A) 20
B) 15
C) 17
D) The series diverges so no sum is possible.
To find the number of terms needed to obtain a sum with an error less than 0.05, we first need to determine if the series converges or diverges.
The given series is the summation from n equals 1 to infinity of (-1)^(n+1)/n. This is an alternating series because the sign changes for each term.
To test the convergence of the series, we can use the alternating series test. For an alternating series to converge, two conditions must be met:
1. The terms of the series approach zero as n approaches infinity.
2. The terms of the series are decreasing.
Let's check these conditions:
1. As n approaches infinity, the absolute value of (-1)^(n+1)/n will approach zero. Hence, this condition is satisfied.
2. To check if the terms are decreasing, we can examine the absolute values of consecutive terms.
The absolute value of the nth term is (-1)^(n+1)/n, and the absolute value of the (n+1)th term is (-1)^((n+1)+1)/(n+1).
Taking the ratio of the absolute values of these terms, we get:
|(-1)^(n+1)/n| / |(-1)^((n+1)+1)/(n+1)|
= |(-1)^(n+1)/n| * |(n+1)/[(-1)^(n+2)]|
= |(-1)^(n+1)/n| * (n+1)
Now, we need to show that (n+1)/(n*|(-1)^(n+1)/n|) is less than or equal to 1 for all n.
Considering n > 0, we have:
(n+1)/(n*|(-1)^(n+1)/n|) <= 1
Simplifying the expression:
(n+1)/(n*|(-1)^(n+1)/n|) = (n+1)/(n*1/n) = (n+1)/n = 1 + 1/n
Since 1 + 1/n is always greater than 1 for any positive integer n, the terms are decreasing.
As both conditions of the alternating series test are met, we conclude that the given series converges.
To determine the number of terms needed to achieve an error less than 0.05, we can use the formula for the remainder of an alternating series:
|R| <= |a(n+1)|
where R is the remainder, a(n+1) is the (n+1)th term, and |a(n+1)| is its absolute value.
In this case, the remainder is the error in the sum when using a finite number of terms.
|R| = |(-1)^(n+2)/(n+1)|
We want |R| to be less than 0.05, so we can set up the inequality:
|(-1)^(n+2)/(n+1)| < 0.05
Now, we can solve this inequality to find the number of terms needed.
Taking the reciprocal of both sides:
1/|(-1)^(n+2)/(n+1)| > 1/0.05
Simplifying the right side:
1/0.05 = 20
|(-1)^(n+2)/(n+1)| > 20
Since (-1)^(n+2) alternates between -1 and 1 for odd and even values of (n+2) respectively, we can rewrite the inequality:
(-1)/(n+1) > 20 or 1/(n+1) > 20
Simplifying, we get two separate inequalities:
1/(n+1) > 20 (for odd values of n)
-1/(n+1) > 20 (for even values of n)
Solving these inequalities, we find:
1/(n+1) > 20 implies n < -19 (which is not possible since n is a positive integer)
-1/(n+1) > 20 implies n < -21 (which is not possible since n is a positive integer)
Hence, there are no values of n that satisfy the inequalities, and the error will never be less than 0.05 regardless of the number of terms added.
Therefore, the correct answer is D) The series diverges, so no sum is possible.