Question

[Mi Tạ Tiểu]

CMR :

\(\dfrac{3}{1^2.2^2}+\dfrac{5}{2^2.3^2}+......+\dfrac{4019}{2009^2.2010^2}< 1\)

Answer 1

[Đạt Trần Tiến]

Ta có:

(n+1)²-n²=2n+1=n+(n+1)

=> A=\(\frac{2+1}{2^21^2}+\frac{2+3}{2^23^2}+... +\frac{2009+2010}{2009^22010^2}=1-\frac{1}{2^2}+\frac{1}{2^2} -\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+...+\frac{1}{2009^2}-\frac{1}{2010^2} <1 \)