Find the sum of the series [infinity] (−1)n n! n = 0 correct to three decimal places. SOLUTION We first observe that the series is convergent by the Alternating Series Test because (i) 1 (n + 1)! = 1 n!(n + 1) Correct: Your answer is correct. 1 n! (ii) 0 < 1 n! ≤ 1 n → 0 Correct: Your answer is correct. so 1 n! → 0 Correct: Your answer is correct. as n → [infinity]. To get a feel for how many terms we need to use in our approximation, let's write out the first few terms of the series: s = 1 0! − 1 1! + 1 2! − 1 3! + 1 4! − 1 5! + 1 6! − 1 7! + ⋯