Question

[le kim ngoc]

So sánh

a) A= 2003/2004+2004/2005+2005/2003 với 3

b) B= 1/2²+1/3²+1/4²+.......+1/2015² với 1

c) C+ 1/2²+1/3²+......+1/100 với 75/100

Answer 1

[Akai Haruma]

a/

$A-3=\frac{2003}{2004}+\frac{2004}{2005}+\frac{2005}{2003}-3$

$=(1-\frac{1}{2004})+(1-\frac{1}{2005})+(1+\frac{2}{2003})-3$

$=\frac{2}{2003}-\frac{1}{2004}-\frac{1}{2005}$

$=(\frac{1}{2003}-\frac{1}{2004})+(\frac{1}{2003}-\frac{1}{2005})$

$>0+0=0$

$\Rightarrow A>3$

Answer 2

[Akai Haruma]

b/

$B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....+\frac{1}{2015^2}$

$< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}$

$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}$

$=1-\frac{1}{2015}<1$