S=1/30+1/31+1/32+1/33+...+1/59+1/60

S có 31 phân số,ta thấy:

1/30>1/62                             1/31>1/62                          1/32>1/62         ............          1/60>1/62

Vậy:

S>31.1/62

S>31/62

S>1/2

Vậy S>1/2

Chúc em học tốt^^

\(S=1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\)

\(\frac{S}{2}=\frac{1}{2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{9\cdot10}\)

\(\frac{S}{2}=\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)

\(\frac{S}{2}=1-\frac{1}{10}\)

\(\frac{S}{2}=\frac{9}{10}\)

\(S=\frac{9}{5}

\(5S=1+\frac{2}{5}+\frac{3}{5^2}+...+\frac{2015}{5^{2014}}\Rightarrow4S=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}-\frac{2015}{5^{2015}}\)

Đặt B = \(1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2014}}\)

 => 5B = \(5+1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{2013}}\)

=> 4B = \(5-\frac{1}{5^{2014}}

Ta có :S=1/11+1/12+1/13+...+1/20<1/10+1/10+1/10+...+1/10(20 số hạng 1/10)

=>S<1/10.20=1/2<5/6

ĐPCM

S>5 . 2⁹⁸

A = 1 + 2 + 2² + 2³ + .... +2⁹⁸

2A = 2(1 + 2 + 2² + 2³ +....+2⁹⁸)

2A = 2 + 2² + 2³ + 2⁴ +....+2⁹⁹

2A - A = (2 + 2² + 2³ + 2⁴ +.....+ 2⁹⁹) - (1 + 2 + 2² + 2³ +.....+2⁹⁸)

A = ( 2 - 2 ) +  ( 2² - 2² ) + (2³ - 2³).....+(2⁹⁸ - 2⁹⁸) + 2⁹⁹ - 1

A = 0 + 0 + 0 +.....+0  + 2⁹⁹ - 1

A = 2⁹⁹ - 1

So sánh : 

2⁹⁹ - 1 và 5 . 2⁹⁸

2⁹⁹ - 1 < 5 . 2⁹⁸

( Biết hay sai thì chịu nhe hehe phần so sánh tui