Question 1
Let $a$ be a nonnegative real number. Define $u_1=3$ and
$$u_{n+1}=\frac12u_n+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3}\qquad(n\ge1).$$
a) For $a=0$, prove that the sequence has a finite limit and find it.
b) For every $a\in[0,1]$, prove that the sequence has a finite limit.
$$u_{n+1}=\frac12u_n+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3}\qquad(n\ge1).$$
a) For $a=0$, prove that the sequence has a finite limit and find it.
b) For every $a\in[0,1]$, prove that the sequence has a finite limit.