Estimate the error in the partial sum S9S_9S9 for the alternating series ∑n=1∞(−1)nn3\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}∑n=1∞n3(−1)n.
∣R9∣≤1103|R_9| \leq \frac{1}{10^3}∣R9∣≤1031
∣R9∣≤193|R_9| \leq \frac{1}{9^3}∣R9∣≤931
∣R9∣≤183|R_9| \leq \frac{1}{8^3}∣R9∣≤831
∣R9∣≤192|R_9| \leq \frac{1}{9^2}∣R9∣≤921