Iteration on sequences I am trying to find a general formula for this sequence, in order to find its limit (has to be this way):
$$X_{k+1} = \frac {X_k} 2 + \frac 1 {X_k}$$
I cannot get a grasp on it; after $2$ or $3$ iterations, the sequence gets huge and I don't see the pattern.
Any help is greatly appreciated!

If the sequence converges, it must converge to an $x$ such that
$$x=\frac x2+\frac1x$$ or $$x^2=2.$$
Then with $x_k:=y_k+\sqrt2$,
$$y_{k+1}+\sqrt 2=\frac{y_k+\sqrt2}2+\frac1{y_k+\sqrt2}$$ and
$$y_{k+1}=\frac{y_k^2}{2(y_k+\sqrt2)}\le\frac{y_k^2}{2\sqrt2}.$$
You easily see that starting from $y_0<1$, you get a (quickly) decreasing sequence with limit $0$.
