Mathematics, 21.11.2019 01:31 taniyahreggienae
Find the sum of the series [infinity] (โ1)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) 1 n! (ii) 0 < 1 n! โค 1 n โ so 1 n! โ as n โ [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! โ 1 1! + 1 2! โ 1 3! + 1 4! โ 1 5! + 1 6! โ 1 7! + โฏ = 1 โ 1 + 1 2 โ 1 6 + 1 24 โ 1 120 + 1 โ 1 5040 + โฏ notice that b7 = 1 5040 < 1 5000 = and s6 = 1 โ 1 + 1 2 โ 1 6 + 1 24 โ 1 120 + 1 720 โ (rounded to six decimal places). by the alternating series estimation theorem we know that |s โ s6| โค b7 โค 0.0002. this error of less than 0.0002 does not affect the third decimal place, so we have s โ 0.368 correct to three decimal places.
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