Closed form for nth term - generating functions I think I am mostly confused about what the question is asking. I read that "closed form" means that it should not be represented as as infinite sum, so I am not sure what they are asking for. Would they like a summation symbol with the summation of all numbers through the nth term?
"For the generating functions below, give a closed form for the nth term of its associated sequence."


*

*$ 3x^4 +7x^3−x^2 +10 + \frac{1}{1-x^3}$

*$(1+x)^{10}$
 A: No: a summation is not a closed form. Roughly speaking, a closed form is a function of $n$ not involving a summation or product whose length depends on $n$.
Here’s a simple example. The generating function $\frac1{1-2x}$ represents the series $$\sum_{n\ge 0}(2x)^n=\sum_{n\ge 0}2^nx^n\;,$$ whose associated sequence of coefficients is $\langle a_0,a_1,a_2,\ldots\rangle=\langle 2^0,2^1,2^2,\ldots\rangle$. A closed form for $a_n$ is $a_n=2^n$.
Added: For the first one your closed form will have several cases – six if you write it as I would. I would begin by expanding $\frac1{1-x^3}$ into its power series. For the second one you’ll want to use the binomial theorem.
A: Roughly speaking, a "closed form" is a formula that requires less work than just doing the computation. This typically means not having summation signs or other "repeat for all..." structures. But in any case, you'd have to agree on some set of "elementary" functions, i.e., powers, factorials, binomial coefficients are normally considered such.
