A question about convergence interval of power series Could you give me some hint how to solve this problem:
Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$.
I managed to prove that $a_n$ is decreasing sequence, $a_n\to0$ and radius of convergence of power series $\sum_{n\ge1}a_nx^n$ is 2. 
This is all quite standard staff, but how to find convergence interval ?
I could not decide about convergence of $\sum_{n\ge1}2^na_n$ because:
1)ratio test is inconclusive;
2)root test=$2\sqrt[n]{a_n}$ and I could not estimate $\sqrt[n]{a_n}$;
3)I tried comparison test,knowing that $\sum_{n\ge1}2^na_{2^n}$  but could not compute the $\lim_{n\to\infty}\frac{a_n}{a_{2^n}}$, $a_n$ is decreasing, therefore from some n$a_n\ge a_{2^n}$ but does $\frac{a_n}{a_{2^n}}$ converge to finite number ?
Thanks.

Hint: Note that
$$a_{n + 1} > \frac 1 2 a_n$$
for every $n$, so a brief argument (perhaps with induction) shows that
$$a_{n + 1} > \frac 1 {2^n} a_n$$
Hence $2^{n + 1} a_{n + 1} > 2 a_n$. Consider something similar at $-2$.
