Câu hỏi:
Cho (S = frac{1}{{{5^2}}} + frac{1}{{{7^2}}} + frac{1}{{{9^2}}} + …. + frac{1}{{{{103}^2}}}). Kết luận nào sau đây đúng?
Lời giải tham khảo:
Đáp án đúng: C
Ta có:
(begin{array}{l} frac{1}{{{5^2}}} < frac{1}{{4.6}};frac{1}{{{5^7}}} < frac{1}{{6.8}};frac{1}{{{9^2}}} < frac{1}{{8.10}};…….;frac{1}{{{{103}^2}}} < frac{1}{{102.104}}\ Rightarrow S = frac{1}{{{5^2}}} + frac{1}{{{7^2}}} + frac{1}{{{9^2}}} + …. + frac{1}{{{{103}^2}}} < frac{1}{{4.6}} + frac{1}{{6.8}} + frac{1}{{8.10}} + …. + frac{1}{{102.104}}\ Rightarrow S < frac{1}{2}left( {frac{1}{4} – frac{1}{6} + frac{1}{6} – frac{1}{8} + frac{1}{8} – frac{1}{{10}} + ….. + frac{1}{{102}} – frac{1}{{104}}} right)\ Rightarrow S < frac{1}{2}left( {frac{1}{4} – frac{1}{{104}}} right)\ Rightarrow S < frac{{25}}{{208}} < frac{{25}}{{160}}\ Rightarrow S < frac{5}{{32}} end{array})
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