Taylor series of polynomial is the same polynomial

Statement

Suppose $f$ is a polynomial with real coefficients. Then, the Taylor series of $f$ about any point $x_0$, viewed as a function, turns out to be the same polynomial $f$.

Proof

Proof about 0

Given: A polynomial $f(x):=\sum _{k=0}^n{a}_k{x}^k$

To prove: The Taylor series of $f$ about 0 is $f$

Proof: For any $k$, the coefficient of $x^k$ in the Taylor series is $\frac{f^{(k)}(0)}{k!}$. We note that:

• If $k>n$, then $f^{(k)}$ is the zero function, and the coefficient of $x^k$ is zero.
• If $0\le k\le n$, then we can easily see that $f^{(k)}(0)=k!a_k$, so $\frac{f^{(k)}(0)}{k!}=a_k$.

Putting the pieces together, we get that the Taylor series is the same polynomial.