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? 

$$a_{0}=0\\a_{n+1}=\frac{a_{n}}{2}+1\\a_{1}=\frac{a_{0}}{2}+1=1\\a_{2}=\frac{a_{2}}{2}+1=1+\frac{1}{2}\\a_{3}=\frac{\frac{3}{2}}{2}+1=1+\frac{3}{4}\\a_{4}=\frac{\frac{7}{4}}{2}+1=1+\frac{7}{8}\\a_{5}=\frac{\frac{15}{8}}{2}+1=1+\frac{15}{16}\\a_{n}=\frac{2^{n-1}-1}{2^{n}}+1\\...\\a_{n}<2\\so\\it-converge
$$
