Ta có : \(\dfrac{1}{2^2}\)<\(\dfrac{1}{1.2}\); \(\dfrac{1}{3^2}\)<\(\dfrac{1}{2.3}\);.....;\(\dfrac{1}{2016^2}\)<\(\dfrac{1}{2015.2016}\)

⇒ A = \(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}\)+...+\(\dfrac{1}{2016^2}\)< \(\dfrac{1}{1.2}\)+\(\dfrac{1}{2.3}\)+...+\(\dfrac{1}{2015.2016}\)

⇒ A = \(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}\)+...+\(\dfrac{1}{2016^2}\) < 1 - \(\dfrac{1}{2016}\)= \(\dfrac{2015}{2016}\) (ĐCPCM)

\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2016^2}+\frac{1}{2017^2}\)

\(A=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{2016.2016}+\frac{1}{2017.2017}\)

Ta thấy \(\frac{1}{2.2}< \frac{1}{1.2};\frac{1}{3.3}< \frac{1}{2.3};\frac{1}{4.4}< \frac{1}{3.4};...;\frac{1}{2016.2016}< \frac{1}{2016.2017};\frac{1}{2017.2017}< \frac{1}{2017.2018}\)

Suy ra \(A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2016.2017}+\frac{1}{2017.2018}\)

Nên \(A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-...+\frac{1}{2017}-\frac{1}{2018}\)

Khi đó \(A< 1-\frac{1}{2018}< 1\)nên A < 1

Suy ra A - 1 < 0

Vậy A - 1 < 0

Câu a:

M = 1/3 - 1/3^2 + 1/3^3 - 1/3^4 + 1/3^5 - 1/3^6 < 1/4

3M = 1 - 1/3 + 1/3^2 - 1/3^3 + 1/3^4 - 1/3^5

3M + M = 3 - 1/3 + 1/3^2 - 1/3^3 + 1/3^4 - 1/3^5 + 1/3 - 1/3^2 + 1/3^3 - 1/3^4 + 1/3^5 - 1/3^6

4M = (1 - 1/3^6) + (-1/3 + 1/3) + (1/3^2 - 1/3^2) + (1/3^4 - 1/3^4) + (1/3^5 - 1/3^5)

4M = 1 - 1/3^6 + 0 + 0+ ..+ 0

M = 1/4 - 1/4.3^6 < 1/4 (đpcm)

4M = 3 - 1/3^6

M = 3/4

A = 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^n < 1

A = 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^n

2A = 1 + 1/2 + 1/2^2+ ..+ 1/2^n-1

2A - A = 1 + 1/2 + 1/2^2+ ..+ 1/2^n-1 - (1/2 + 1/2^2 + 1/2^3 + ... + 1/2^n)

A = (1 - 1/2^n) + (1/2 - 1/2) + ..+ (1/2^n-1 -1/2^n-1)

A = 1 - 1/2^n

A < 1 (đpcm)

Câu b:

B = 1/3 + 1/3^2 + 1/3^3 + ...+ 1/3^n < 1/2

3B = 1 + 1/3 + 1/3^2 + ..+ 1/3^n - 1

3B - B = 1 + 1/3 + 1/3^2 + ..+ 1/3^n - 1 - (1/3 + 1/3^2 + 1/3^3 + ...+ 1/3^n)

2B = 1 + 1/3 + 1/3^2 + ..+ 1/3^n - 1 - 1/3 - 1/3^2 - 1/3^3 -..- 1/3^n-1 - 1/3^n

2B = (1 - 1/3^n) + (1/3 - 1/3) +..+(1/3^n-1-1/3^n-1)

2B = 1 - 1/3^n

B = 1/2 - 1/2.3^n < 1/2 (đpcm)

\(A< \frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+....+\frac{1}{99.100}\)

\(A< \frac{1}{4}-\frac{1}{100}\)

\(A< \frac{6}{25}< \frac{1}{4}\)

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