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Question Number 201495 by Rodier97 last updated on 07/Dec/23

$$$$$$$$$$$$$$$$$${\left(Un\right)}_{n⩾1;}\frac{1}{{nC}_{2n}^n}$$$$$$$$studyconvergence$$$$$$$$$$$$$$

Commented by Faetmaaa last updated on 07/Dec/23

$$\mathrm{\forall }n⩾10⩽Un⩽\frac{1}{n}$$$$\mathrm{So}\lim {\left(Un\right)}_{n⩾1}=0$$

Answered by MM42 last updated on 07/Dec/23

$$u_n=\frac{{(n!)}^2}{n\times \left(2n\right)!};u_n>0\left(i\right)$$$$u_{n+1}=\frac{{((n+1)!)}^2}{(n+1)\times (2n+2)!}$$$$$$$$\frac{u_{n+1}}{u_n}=\frac{{((n+1)!)}^2}{(n+1)\times (2n+2)!}\times \frac{n\times \left(2n\right)!}{{\left((n!)\right)}^2}$$$$=\frac{{(n+1)}^2\times {\left((n!)\right)}^2}{2{(n+1)}^2\times (2n+1)\left(2n\right)!}\times \frac{n\times \left(2n\right)!}{{\left((n!)\right)}^2}$$$$=\frac{n}{2(2n+1)}<1$$$$\Rightarrow u_{n}isdecrasing\left(ii\right)$$$$\left(i\right),\left(ii\right)\to u_{n}isvonvergence$$$$$$

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