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 is convergent. Use the Alternating Series Test to find an upper bound on the absolute error if the 7th partial sum is used as an estimate of the infinite sum.
Type your answer in the space below and give three decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482). (10 points)
Please Explain!!!
To determine if a series is convergent using the Alternating Series Test, we need to check two conditions:
1. The terms in the series alternate signs.
2. The absolute value of each term in the series is decreasing.
Let's analyze the series step by step:
The given series is:
∑((-1)^(n+1) / n), where n starts from 1 and goes to infinity.
Step 1: Check if the series alternates signs:
The terms in the series alternate signs because we have (-1)^(n+1), where n+1 is an odd number, which means the exponent is odd, resulting in alternation between positive and negative signs.
Step 2: Check if the absolute value of each term is decreasing:
To check for the decreasing absolute value of each term, let's examine the ratio of each term to the next term:
|((-1)^(n+2) / (n+1)) / ((-1)^(n+1) / n)|
Simplifying the expression above, we get:
|((-1)^(n+2) * n) / ((-1)^(n+1) * (n+1))|
The absolute value of this expression is equal to:
|(-1 * n) / ((-1) * (n+1))| = |n / (n+1)| = n / (n+1)
Now, let's consider the function f(x) = x / (x+1). We can see that this function is a decreasing function for x >= 1.
Step 3: Since the series alternates signs and the absolute values of the terms are decreasing, we can conclude that the given series is convergent.
Now, to find an upper bound on the absolute error if the 7th partial sum is used as an estimate of the infinite sum, we need to use the Alternating Series Error Bound formula:
Error ≤ | R_(n+1) | , where R_(n+1) is the remainder after the nth term.
The remainder term for an alternating series can be given by:
R_(n+1) = | a_(n+1) |, where a_(n+1) is the (n+1)th term.
In this case, the 7th partial sum will be the sum of the first 7 terms:
S_7 = ∑((-1)^(n+1) / n) from n=1 to 7
Calculating S_7, we get:
S_7 = (-1/1) + (1/2) + (-1/3) + (1/4) + (-1/5) + (1/6) + (-1/7) = -0.960
Now, let's find the 8th term:
a_8 = (-1)^(8+1) / 8 = (-1)^9 / 8 = -1/8
Finally, calculate the absolute error using the Alternating Series Error Bound formula:
Error ≤ | R_8 | = | a_8 | = |-1/8| = 1/8 = 0.125 (to three decimal places)
Therefore, the upper bound on the absolute error is 0.125