Để ý:
$a+b=a^2-b^2$ với $a=b+1$và $a, b \in N$
Áp dụng:
$B=\dfrac{3}{(1.2)^2}+\dfrac{5}{(2.3)^2}+\dfrac{7}{(3.4)^2}+...+\dfrac{4029}{(2014.2015)^2}$
$\iff B=\dfrac{2^2-1^2}{(1.2)^2}+\dfrac{3^2-2^2}{(2.3)^2}+\dfrac{4^2-3^2}{(3.4)^2}+...+\dfrac{2015^2-2014^2}{(2014.2015)^2}$
$\iff B=\dfrac{2^2}{1^2.2^2}-\dfrac{1^2}{1^2.2^2}+\dfrac{3^2}{2^2.3^2}-\dfrac{2^2}{2^2.3^2}+...+\dfrac{2015^2}{2014^2.2015^2}-\dfrac{2014^2}{2014^2.2015^2}$
$\iff B=1-\dfrac{1}{2^2}+\dfrac{1}{2^2 }-\dfrac{1}{3^2}+...+\dfrac{1}{2014^2}-\dfrac{1}{2015^2}$
$\iff B=1-\dfrac{1}{2015^2}$
$\iff B=\dfrac{4060224}{4060225}$
Bến dưới có nút "Đúng", có gì bạn bấm vào giùm =))