(A) Consider the following alternating series: ?_{n=1}^{?} rac{(-1)^{n-1} ?n}{n^3 + 1}. (i) Show that the alternating series is absolutely convergent. (ii) Estimate the error involved in approximating the sum of the alternating series ?_{n=1}^{?} rac{(-1)^{n-1} ?n}{n^3 + 1} using the third partial sum S3. Round your answer to four places of decimals. (B) Use the Integral Test to determine whether the series ?_{n=2}^{?} rac{?n n}{n} converges or diverges after showing that the conditions of the Integral Test are satisfied.
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(A) Consider the following alternating series: Show that the alternating series is absolutely convergent. Estimate the error involved in approximating the sum of the alternating series using the third partial sum S3. Round your answer to four decimal places. (B) Use the Integral Test to determine whether the series converges or diverges after showing that the conditions of the Integral Test are satisfied.
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