u-1-1-u-n-1-n-2-n-1-n-2-n-R-lim-n-u-n-

Question Number 138571 by bemath last updated on 15/Apr/21

{\begin{cases} u_{1} = 1 \\ u_{n + 1} = \frac{n^{2} - n + 1}{n^{2}} \end{cases}; \forall n \in R

\lim_{n \to \infty} u_{n} = ?

Answered by mathmax by abdo last updated on 15/Apr/21

\Rightarrow u_{n} = \frac{{(n - 1)}^{2} - (n - 1) + 1}{{(n - 1)}^{2}} = \frac{n^{2} - 2 n + 1 - n + 2}{n^{2} - 2 n + 1} = \frac{n^{2} - 3 n + 3}{n^{2} - 2 n + 1}

\Rightarrow \lim_{n \to + \infty} u_{n} = \lim_{n \to + \infty} \frac{n^{2}}{n^{2}} = 1