# How is the generating function related to its sequence?

Assume that the sequence $(a_n)^{\infty}_{n=0}$ is given by $a_0=2, a_1=3, a_2=-2, a_3=1, a_4=-3, \text{ and } a_n=0 \text{ for } n \ge 5$
Let $F$ be the generating function for the sequence $(a_n)^{\infty}_{n=1}$. Determine $F(2)$.

The generating function for the sequence $2, 3, -2, 1, -3, 0, 0, 0, \dots$ is $F(x)=2+3x-2x^2+x^3-3x^4$
Then I need to find $F(2)$, this is just substituting $2$ for $x$ in the $F(x)$ right? This gives me $F(2)=-40$

But what is the meaning of $F(2)$? Taylor series was something I did not fully understand when I took calculus and now I am a little confused.

So my question is how is the generating function is related to the sequence and what the meaning of $F(2)$ is for these types of questions? I believe that to find the number in the sequence at position $n$ I would have to take the $n^{th}$ derivative of the generating function $F(x)$ and plug in $x=0$, does this make sense?

Q: Then I need to find $F(2)$, this is just substituting 2 for $x$ in the $F(x)$ right? This gives me $F(2)=−40$.

A: That's right.

Q: But what is the meaning of $F(2)$?

A: Without greater context, I'm not sure if there is a special meaning for $F(2)$. However, the generating function does have special meaning at other values. For instance, $F(0)$ is the initial term of the sequence, and $F(1)$ is the sum of the terms of the sequence.

Q: So my question is how is the generating function is related to the sequence and what the meaning of $F(2)$ is for these types of questions?

A: The generating function is a representation of the entire sequence. The hope is that if you have a good representation of the generating function, then you can calculate (or at least approximate) anything about the underlying sequence.

Q: I believe that to find the number in the sequence at position $n$ I would have to take the $n$th derivative of the generating function $F(x)$ and plug in $x=0$, does this make sense?

A: That is definitely one way to compute a term in the sequence. However, in practice this method rarely works since iterated differentiation often gets unwieldly.