Construction of a special power series

Question

If there exists a power series $\displaystyle\sum_{n=0}^{+\infty}a_nx^n$ whose radius of convergence is 1, and $\forall k\in \mathbb {N}, \displaystyle\sum_{n=0}^{+\infty}(a_nx^n)^{(k)} $ (namely the k-th term-by-term derivation) converges at the point $x=1$?
Furthermore, if the above eample exists, dose that mean for this example the convergence property is also satisfied with $x=-1$?