Convergence of partial sums to a formal power series

Question

I have the following question regarding formal power series.

Consider we have the sequence of formal power series $F_k(x) = \sum_{n}a_n^{(k)}x^{n}$ over a commutative ring R, which converges to $G(x) = \sum_{n}b_{n}x^{n}$ if for each $n$ of the sequence $(a_{n}^{(1)},a_{n}^{(2)},...)$ converges to $b_n$ in the discrete topology. I need to prove that the partial sums $a_0+a_1x+a_2x^2+...a_nx^{n}$ converges to $\sum_na_nx^{n}$.

Could anyone give me a hint on how to work with this kind of problems? I have a very vague idea of how to work with formal power series and so I'm not sure how to even start this problem. Any hint or suggested reading would be greatly appreciated.

Thanks!

Answer 1

for each $n$ of the sequence $(a_{n}^{(1)},a_{n}^{(2)},...)$ converges to $b_n$ in the discrete topology.

This simply means that the sequence $(a_{n}^{(1)},a_{n}^{(2)},...)$ eventually stabilizes at $b_n$. :-)

I need to prove that the partial sums $a_0+a_1x+a_2x^2+...a_nx^{n}$ converges to $\sum_na_nx^{n}$.

And now this is clear, since for this sequence we have $(a_{n}^{(1)},a_{n}^{(2)},\dots)= (0,0,\dots,0, a_n,a_n,\dots)$.

Could anyone give me a hint on how to work with this kind of problems? I have a very vague idea of how to work with formal power series and so I'm not sure how to even start this problem. Any hint or suggested reading would be greatly appreciated.

This problem is trivial, so there is no need to read anything, if suffices only to understand what is essentially asked.