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For an alternating series whose summands are decreasing in magnitude, the true sum S lies between any two successive partial sums:

min{SN, SN+1}Consider S = ∑[infinity] n=1 [(-1)ⁿ⁺¹/n³]
and write Sn = ∑N n=1 [(-1)ⁿ⁺¹/n³]
(a) Find the smallest value of N for which the interval bracketing S in line (*) above has length at most 10-⁻⁵.

1 Answer

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Final answer:

To find the smallest N such that the error between partial sums SN and SN+1 of the alternating series is at most 10^-5, one must find N where 1/(N+1)^3 ≤ 10^-5. After determining the cube root, N must be more than the cube root of 10^5 and also be an integer.

Step-by-step explanation:

The student is asking about the convergence of an alternating series and determining the smallest value of N such that the difference between SN and SN+1 is at most 10^-5. The series in question is S = ∑∞ n=1 [(-1)^n+1/n^3], and the partial sum Sn is ∑N n=1 [(-1)^n+1/n^3]. To find the smallest N that meets the criteria, we must analyze the behavior of the series' terms.

Considering the terms decrease in magnitude and alternate in sign, the error introduced by stopping at a particular partial sum SN rather than continuing to infinity is less than or equal to the first omitted term in magnitude. In other words, the value of |SN+1 - SN| = 1/(N+1)^3 must be at most 10^-5 to satisfy the condition. Thus, we need to find the smallest N such that 1/(N+1)^3 ≤ 10^-5.

To solve for N, we can take the cube root of both sides of the inequality:

(N+1) ≥ (10^5)^(1/3)

Through calculation, we find that (10^5)^(1/3) is approximately which implies that N must be greater than this value. After finding the cube root and ensuring that N is a whole number (since the series is defined with natural numbers n), one can determine the smallest N that meets the criteria.

This question illustrates the application of knowledge about series expansions and alternating series, specifically relating it to the rule that as the number N decreases in magnitude, the next term's exponent increases, which affects the convergence and accuracy of the series.

User Sarim Sidd
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