Does the sequence converge, and to what? We have a sequence $\{a_n\}$ 

$$a_0 = 0$$

$$a_{n+1} = \frac{a_{n}}{2} + 1$$

Does it converge? And to what?
Hint: show by induction that the sequence $a_n$ increase. Trick to do this is to see that

$a_{n+1}-a_n=a_n/2-a_{n-1}/2$.

Now this shows that the sequence is increasing. What else do you need to see it converges?

Further, suppose that the limit is $L$. Then $L=L/2+1$, as $\lim a_{n+1}=\lim a_n $
$$a_{0}=0\\a_{n+1}=\frac{a_{n}}{2}+1\\a_{1}=\frac{a_{0}}{2}+1=1\\a_{2}=\frac{a_{2}}{2}+1=1+\frac{1}{2}\\a_{3}=\frac{\frac{3}{2}}{2}+1=1+\frac{3}{4}\\a_{4}=\frac{\frac{7}{4}}{2}+1=1+\frac{7}{8}\\a_{5}=\frac{\frac{15}{8}}{2}+1=1+\frac{15}{16}\\a_{n}=\frac{2^{n-1}-1}{2^{n}}+1\\...\\a_{n}&lt;2\\so\\it-converge
$$