My lecture notes say:
A polynomial on $\mathbb{R}$ is a function $f: \mathbb{R} \to \mathbb{R}$ with $x \mapsto \sum_{k=0}^n a_k x^k$ with $n \in \mathbb{N}$ and $a_k \in \mathbb{R}$.
A power series in $x \in \mathbb{R}$ is the expression $\sum_{k=0}^\infty a_k x^k$ with $a_k \in \mathbb{R}$.
I don't really understand the difference here. What should "expression" mean in this case? Except that the second sum goes to infinity the definitions look the same to me.
Can somebody explain me the differences?
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$\begingroup$The formal polynomial expression is a function because you can always substitute a numerical value for $x$ and get a number. That's not necessarily the case for a power series, since there may well be values for $x$ for which the infinite series does not converge. When restricted to the values of $x$ for which it does converge, it defines a function.
That's why your instructor was careful to call the formal power series just an expression.
Related, possibly helpful: What actually is a polynomial?
$\endgroup$ 2 $\begingroup$It's an expression rather than a function because the series might not converge anywhere except at $x=0$.
$\endgroup$ $\begingroup$There's, strictly speaking, a difference between a polynomial and a polynomial function. Polynomials are expressions also (look up polynomial rings). But when you're dealing with a polynomial ring over an infinite field, then there's a natural bijection between the polynomials and the polynomial functions. Since $\mathbb{R}$ is an infinite field, this distinction is often not made. A power series is an expression just like polynomials are an expression (to not be understood as a function). However, with the topological structure on $\mathbb{R}$, you can use convergence (either pointwise or compact convergence) of functions to assign a power-series to a function defined on some interval of $\mathbb{R}$.
$\endgroup$ 4 $\begingroup$In short terms polynomials are a kind of (regular and smooth) functions, whereas power series can converges (in a given domain) to others functions or not converge at all.
EG
$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}...$$
$\endgroup$ $\begingroup$Polynomials can be defined in this way:
A polynomial is a power series such that there is $k_0 \in \mathbb N_0$ for which we have $a_k=0$ for every $k \geq k_o$.
Now you can see that polynomials can be seen as a special subset of a set of all power series, such that only a finite number of coefficients is non-zero.
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