已知 $$ex x_n=sum_{i=1}^n frac{1}{i(i+1)(i+2)(i+3)}. eex$$ 试证: $sed{x_n}$ 收敛, 并求其极限.
证明: $$eex ea x_n&=frac{1}{3}sum_{i=1}^n frac{(i+3)-i}{i(i+1)(i+2)(i+3)}\ &=frac{1}{3}sez{ sum_{i=1}^n frac{1}{i(i+1)(i+2)} -sum_{i=1}^n frac{1}{(i+1)(i+2)(i+3)} }\ &=frac{1}{3}sez{frac{1}{1cdot 2cdot 3} -frac{1}{(n+1)(n+2)(n+3)}}\ & o frac{1}{18}quadsex{n oinfty}. eea eeex$$