Question

lim(1/1·3 + 1/3·5 + ... + 1/(2n-1)·(2n+1))
x—›беск​

Answer 1

$\dfrac{1}{1\cdot 3}+\dfrac{1}{3\cdot 5}+\dfrac{1}{(2n-1)(2n+1)}=\dfrac{1}{2}\cdot \Big(\dfrac{3-1}{1\cdot 3}+\dfrac{5-3}{3\cdot 5}+...+\dfrac{2n+1-(2n-1)}{(2n-1)(2n+1)}\Big)\\ \\ \\=\dfrac{1}{2}\cdot \Big(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\Big)=\dfrac{1}{2}\cdot \Big(1-\dfrac{1}{2n+1}\Big)=\dfrac{1}{2},~ x\to \infty$