How can I construct a power series from any given polynomial? Suppose I have some polynomial:
$$f(x)=a_0 +a_1x+a_2 x^2+a_3 x^6+a_4x^8$$
How can I construct a power series representation for this polynomial? More generally: How can I construct a power series given ANY polynomial with finite terms?

A (real or complex) polynomial in a single variable $x$
$$ 
 f(x) = a_0 + a_1 x + \ldots + a_n x^n
$$
it equal to its own power series at $x=0$: With
$$
 c_j = \begin{cases}
a_j & \text{for } 0 \le j \le n\\
0 & \text{for }  j > n
\end{cases}
$$
we have 
$$
 f(x) = \sum_{j=0}^n a_j x^j = \sum_{j=0}^\infty c_j x^j \, .
$$
The series converges for all $x \in \Bbb R$ (or $\Bbb C$), so that the radius of convergence is $\infty$.
For the power series centered at an arbitrary point $x_0$ we have
$$
 f(x) = \sum_{j=0}^n\frac{f^{(j)}(x_0)}{j!} x^j = \sum_{j=0}^\infty \frac{f^{(j)}(x_0)}{j!} x^j
$$
because $f^{(j)} \equiv 0$ for $j > n$.
