Сходимост на ред

[Гост]

Изследвайте сходящ ли е реда [tex]\sum_{n=1}^{\infty }\frac{n!(2n)!}{(3n)!}[/tex]

[Добромир Глухаров]

$u_n=\frac{n!(2n)!}{(3n)!}=\frac{n!}{(2n+1)(2n+2)...(2n+n)}<\frac{n^n}{(2n+1)(2n+2)...(2n+n)}=$

$=\frac{1}{(2+\frac{1}{n})(2+\frac{2}{n})(2+\frac{3}{n})...(2+1)}<\frac{1}{2^n}\to0$

$\frac{u_{n+1}}{u_n}=\frac{\frac{(n+1)!(2n+2)!}{(3n+3)!}}{\frac{n!(2n)!}{(3n)!}}=\frac{(n+1)(2n+1)(2n+2)}{(3n+1)(3n+2)(3n+3)}\to\frac{1.2.2}{3.3.3}=\frac{4}{27}<1$

$\Rightarrow$ редът е сходящ.