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.