A= 1+1/1.2 + 1/1.2.3 +......+1/1.2.3.n <1+1/1.2 +1/1.3+......+1/k(k+1)
Những câu hỏi liên quan
A=1+1/1.2+1+1/1.2.3+.......+1/1.2.3.n < 1+1/1.2+1/2.3+.......+1/k(k+1)
Cmr : 1 + 1/1.2 + 1/1.2.3 + .....+ 1/1.2.3....n < 2
Nhận thấy 1/1.2.3 = 1/2.3; 1/1.2.3.4 < 1/3.4; 1/1.2.3.4.5 < 1/4.5; 1/1.2.3...n < 1/n(n-1)
=> 1 + 1/1.2 + 1/1.2.3 +... + 1/1.2.3...n < 1 + 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/n(n-1)
=> 1 + 1/1.2 + 1/1.2.3 +... + 1/1.2.3...n < 1 + 1 -1/2 + 1/2 - 1/3 + 1/3 - 1/4 +...+ 1/n-1 - 1/n
=>1 + 1/1.2 + 1/1.2.3 +... + 1/1.2.3...n < 2 - 1/n < 2
=> đpcm
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Cmr : 1 + 1/1.2 + 1/1.2.3 + .....+ 1/1.2.3....n < 2
Cmr : 1 + 1/1.2 + 1/1.2.3 + .....+ 1/1.2.3....n < 2
A=\(\left(1-\dfrac{1}{1.2}\right)\left(1-\dfrac{1}{1.2.3}\right).........\left(1-\dfrac{1}{1.2.3....n}\right)\)
CM:
1+\(\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3...n}< 2\)
Đặt A = \(1+\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3....n}\)
Ta có: \(\frac{1}{1.2}=\frac{1}{1.2}\)
\(\frac{1}{1.2.3}=\frac{1}{2.3}\)
\(\frac{1}{1.2.3.4}< \frac{1}{3.4}\)
..............
\(\frac{1}{1.2.3....n}< \frac{1}{\left(n-1\right)n}\)
Cộng vế với vế ta được:
\(A< 1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}=1+1-\frac{1}{n}=2-\frac{1}{n}< 2\)(đpcm)
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Chứng minh bất đẳng thức sau:
\(1+\dfrac{1}{1.2}+\dfrac{1}{1.2.3}+...+\dfrac{1}{1.2.3.....n}< 2\)
Chứng tỏ: 1/1.2+1/1.2.3+1/1.2.3.4+...+1/1.2.3...100<1
Tính nhanh: 1+1.2+1.2.3+1.2.3.4+...+1.2.3...99+1.2.3....100
tính tổng dãy số thì dễ nhưng hãy viết rõ ràng hơn
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