Convergence of formal power series

Question

So I've just come across formal power series and was somewhat interested, but still can't seems to understand them. I was reading this explanation about the convergence of a formal power series and was wondering about the convergence of the following formal power series:

\begin{align} 1+x+x^2+x^3+\cdots \\ \\ 1+2x+3x^2+4x^3+\cdots \\ \\ 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \end{align}

From what I understand after reading this, the first formal power series is convergent because the coefficient is stable, but what about the second and third formal power series?

Is there an example where the normal power series is convergent for some value of $x$, but the corresponding formal power series is not convergent?

Thanks!

Answer 1

A formal power series in the ring $R$ is a sequence $$ (a_0, a_1, \ldots ), $$ where the $a_i$ lie in $R$, although this is more often denoted by $$ \sum_{n=0}^\infty a_nX^n. $$ Nevertheless, all of the symbols you see above, except for the coefficients $a_i$ themselves, are just superfluous decorations.

The set of all formal power series is denoted by $R[[X]]$ and it has the structure of a ring, where you add two formal power series just by adding their respective coefficients, while the multiplication is defined by $$ \Big (\sum_{n=0}^\infty a_nX^n\Big ) \Big (\sum_{n=0}^\infty b_nX^n\Big )= \Big (\sum_{n=0}^\infty c_nX^n\Big ), $$ where $$ c_n:= \sum_{k=0}^n a_k b_{n-k}. $$

In addition, $R[[X]]$ is a topological ring by identifying it with the product topological space $R^{\mathbb N}$ in the obvious way, where $R$ is given the discrete topology.

This means that a sequence $$ \Big\{\sum_{n=0}^\infty a_{n, k}X^n\Big\}_k $$ converges to an element $\sum_{n=0}^\infty a_nX^n$ in $R[[X]]$, if and only for every $n$, there is a $k_0$, such that $a_{n,k}= a_n$, for every $k\geq k_0$. In others words, if you focus on the coefficients of a single $X^n$, you'll notice they stop changing after a while.

Whenever you have a topological ring, you can make sense of convergence of "infinite sums", such as, $\sum_{n=0}^\infty P_n$, by simply interpreting it as the sequence of partial sums $$ \Big\{\sum_{n=0}^N P_n\Big\}_N. $$

In the case of $R[[X]]$, one may choose $P_n: = a_nX^n$ as a formal power series in which all coefficients, except for $a_n$, are taken to be zero. In that case you may ask whether or not the "infinite sum" $$ \sum_{n=0}^\infty a_nX^n. $$ converges in $R[[X]]$. Besides observing that the above is NOT to be understood as an element of $R[[X]]$ (it is an infinite sum of elements of $R[[X]]$), it is not hard to see that it does converge (Exercise: prove it!) to the element of $R[[X]]$ you might write as $(a_0, a_1, \ldots )$ in its most formal presentation.

This might be expressed by saying that every power series converges in the ring of formal power series, regardless of how big the coefficients are!