All terms of this sequence are equal to 1: $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$

sequences-and-series elementary-number-theory reference-request recurrence-relations

It does not. The first term (whether $n=0$ or $n=1$) could be $-1$ and all others would be $1$

Related

Is this an integer?
What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?
Pedagogical examples of distinguishing "types of symmetry"
Prove that the given polynomial is irreducible in $\mathbb{Z}[X]$.
A problem of central simple algebras: why $(E,s,\gamma)\cong M_n(F)$ only if $\gamma$ is the norm of an element of $E$?
Between any two continuous functions $f>g$, can we find a real-analytic function?
how to place a rope with a given length within an orthogonal triangle (see picture)
Singular points of orders of a number field
A determinant involving a polynomial is $0$
Finding a metric to make a certain curve a circle
Lorenz Attractor, its Geometric Model, and 14th Smale's problem.
Statements on the behavior of solutions to $y' = \sin(xy)$ for large $y(0)$