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 $f$.

Related facts

Similar facts

• Composite of Taylor series operator and power series summation operator is identity map

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 $0\leq k\leq n$, 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.