What is the technical difference between a formal and informal power series? In my lecture notes the professor wrote that $$e^x = \Sigma \frac{x^k}{k!}$$
is a formal power series because we can plug in whatever we want in $x$ and both side will equate
This is an obvious conclusion, I wonder what he means by formal power series and when is a power series considered informal.
 A: A formal power series is a power series where the question of convergence is not considered.  So a formal power series is just a sequence with a funny notation.  We can write down things like $$\sum_{k=0}^\infty k!x^k.$$ This series doesn't converge anywhere except at $x = 0$, but it's perfectly well defined as a formal power series.
I think you would have to ask your professor precisely what he means in this context.  It doesn't really make sense to say things like "this power series is a formal power series because it has such and such properties...".  To say it's a formal power series just means that we are going to write down the symbols $\sum_{n=0}^\infty a_n x^n$ whether it converges or not.  Strictly speaking, I think you would define all power series to be formal power series, and then for any particular power series you can ask whether it converges anywhere or not.
A: A formal power series is a power series where you don't care about convergence (or even really what "$x$" is). For example, $$1+x+x^2+x^3+\dots$$ is a formal power series, including for $x\geq 1$ (where it wouldn't converge) or $x$ not even defined! You can think of it as just an infinite tuple, the $x^n$ are just placeholders.
You'll see this a lot in math. "Formal" usually means symbol pushing. For another related example in some special equations called SPDEs, you'll have "formal" definitions. For example, KPZ is "formally defined" as $\partial_t u=\partial_{xx} u+(\partial_x u)^2-\infty+\xi$. Obviously "$-\infty$" isn't really what's going on, the reality is much more subtle. But we may "formally" write down a bunch of symbols.
No one says "informal", at least not that I have heard of. Formal=writing down the symbols without really caring what they mean, if the object exists, etc.
Edit: for a related example, you can talk about "formal" derivatives. For example, if you are in some weird ring with no concepts of differentiability, you may write the "formal derivative" of $x^n$ as $nx^{n-1}$. For example if you're in $\mathbb{Z}/4$, the derivative of $2x^2$ is $4x=0$.
A: An expression of the form $a_0+a_1x+a_2x^2+a_3x^3+ \cdots$, where the $a_n$ are arbitrary real (or complex) numbers, might not have a meaningful value except in the very limited case $x=0$ (when of course it equals $a_0$). For example, consider $\sum_{n=0}^\infty n!x^n.$ nevertheless, one can perform formal algebraic operations on such series, such as addition and multiplication, to produce other such series, by the same rules that apply for series that converge for a nontrivial range of the $x$ variable. These formal series cannot be regarded as functions, like the series for (e.g.) e$^x$. However, they can be given a meaningful mathematical definition as sequences: $\sum_{n=0}^\infty a_nx^n$ is identified with the sequence $(a_0,a_1,...)$, and addition and multiplication are performed on these sequences just as for power series. Thus $$(a_n)_{n\in\Bbb N}+(b_n)_{n\in\Bbb N}=(a_n+b_n)_{n\in\Bbb N}$$ and $$(a_n)_{n\in\Bbb N}(b_n)_{n\in\Bbb N}=\left( \sum_{j=0}^na_jb_{n-j}\right)_{\!n\in\Bbb N}.$$
