How can I construct a power series from any given polynomial?

Question

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?

Answer 1

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 $j > n$.