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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 difference is simply that a polynomial has a largest power of x while a power series can have all positive integers as powers. $\endgroup$ – user247327 Dec 14 '17 at 21:40
  • $\begingroup$ I upvote you now that I understand...but stackexchange won't let me do it...I agree with your observations so my comment will upvote you in principle $\endgroup$ – Sedumjoy Sep 19 '18 at 20:57
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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?

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  • $\begingroup$ "When restricted to the values of $x$ for which it does converge, it defines a function." Yes, but that function doesn't determine the series, if $0$ is the only value of $x$ for which it converges. $\endgroup$ – Robert Israel Dec 14 '17 at 21:29
  • $\begingroup$ @RobertIsrael True, but a power series converging only at $0$ does define a function with domain $\{0\}$. $\endgroup$ – Ethan Bolker Dec 14 '17 at 21:36
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It's an expression rather than a function because the series might not converge anywhere except at $x=0$.

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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 $\mathbb{R}$.

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  • $\begingroup$ The distinction between polynomial and a polynomial function is absolutely correct, however in this context I think we are talking about polynomial as function. $\endgroup$ – user Dec 14 '17 at 21:42
  • $\begingroup$ The OP said he didn't understand the difference. Extra context sometimes helps. $\endgroup$ – coffeecaracal Dec 14 '17 at 21:46
  • $\begingroup$ I've pointed out this only because maybe he is looking for something simpler. By the way your answer is absolutely ok! $\endgroup$ – user Dec 14 '17 at 21:50
  • $\begingroup$ That's a bit assumptive. If the OP wants clarification, that's what the comment sections are for. Also, answers on MSE do not have to appease the OP, they just have to answer the question. They are also meant for future readers. My answer might not be sensible to someone now, but it could be what someone in the future needs. $\endgroup$ – coffeecaracal Dec 14 '17 at 21:53
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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

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}...$$

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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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