chứng tỏ rằng: S=1/1!+1/2!+1/3!+….+1/2019!<3

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chứng tỏ rằng: S=1/1!+1/2!+1/3!+….+1/2019!<3

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Claire 3 tháng 2021-09-26T08:06:19+00:00 1 Answers 3 views 0

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    2021-09-26T08:08:15+00:00

    `#Kenshiro`

    `S = 1/(1!) + 1/(2!) + 1/(3!) + … + 1/(2019!) < 3`

    Ta có : 

    `1/(2!) = 1/1.2`

    `1/(3!) = 1/2.3`

    `1/(2019!) = 1/2019.2020`

    `⇔ 1/1.2 + 1/2.3 + … + 1/2019.2020`

    `⇔ 1 – 1/2 + 1/2 – 1/3 + … + 1/2019 – 1/2020`

    `⇔ 1 – 1/2020 = 2019/2020`

    `⇒ 1 + 1/(1!) + 1/(2!) + 1/(3!) + … + 1/2020 < 1 ⇔ 1 + 1/(1!) + 1/(2!) + 1/(3!) + … + 1/2019 < 3`

    `⇒ S < 3`

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