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)
A) 20
B) 15
C) 17
D) The series diverges so no sum is possible.
ever think of actually writing the math?
sum(n=1..infinity) (-1)^(n+1)/n
or even better,
∑(n=1..∞) (-1)^(n+1)/n
You know that if S_n = ∑(-1)^n a_n then
|S - Sn| < a_(n+1)
so you just need n such that
1/(n+1) < 1/20
n+1 > 20
n > 19
Check:
∑k=1..n for n= 18,19,20 is
0.6661398
0.7187714
0.6687714
S∞ = log2 = 0.69315
you can read online about the alternating harmonic series
To find the number of terms we need to add in order to ensure that the sum has an absolute error less than 0.05, we can use the concept of the Alternating Series Estimation Theorem.
The given series is an alternating series where the terms alternate in sign. The alternating series error bound is given by:
|En| ≤ |Rn+1|
where En is the absolute error in the approximation obtained by summing the first n terms and Rn+1 is the remainder term after summing the first n terms.
In this case, the remainder term Rn+1 is given by:
Rn+1 = (absolute value of the (n+1)th term)
Let's find the number of terms needed by evaluating the remainder term until it becomes less than 0.05.
|Rn+1| = |-1^(n+2)/(n+1)|
To make |Rn+1| < 0.05, we can solve the inequality:
|-1^(n+2)/(n+1)| < 0.05
Upon evaluating different values of n, we find that when n = 16, the remainder term becomes less than 0.05. Therefore, we need to add at least 17 terms to ensure that the sum has an absolute error less than 0.05.
Hence, the correct answer is:
C) 17
To determine the number of terms we need to add in the series to find the sum with an absolute error less than 0.05, we can use the concept of the error bound for alternating series.
The given series is an alternating series because it consists of alternating signs between the terms. Let's denote the terms of the series as aₙ, where aₙ = (-1)^(n+1) / n.
The error bound for an alternating series is given by the absolute value of the first neglected term. In other words, if we stop adding terms after a certain number, the sum of the remaining terms that we didn't include will be less than or equal to the absolute value of the first neglected term.
So, we need to find the value of n that makes |aₙ₊₁| < 0.05.
|aₙ₊₁| = |-1^(n+1+1) / (n+1)|
= 1 / (n+1)
Now, we set 1 / (n+1) < 0.05 and solve for n.
1 / (n+1) < 0.05
1 < 0.05(n+1)
1 < 0.05n + 0.05
0.95 < 0.05n
n > 0.95 / 0.05
n > 19
Therefore, we need to add at least 20 terms in order to ensure that the absolute error of the sum is less than 0.05.
The answer is A) 20.