Let $p(x) = a_kx^k + a_{k-1}x^{k-1} + ... + a_1x+a_0$ be an arbitrary polynomial.
Find the Taylor Series at $0$ for $p$.
Intuitively, I know that the Taylor series is just the polynomial itself $\sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}(x)^k$. Which is easy to see by taking the derivatives of the terms in the polynomial. However, I am trying to prove this with a rigid method, if I have to show it rigidly I assume that I have to prove it by induction:
Prove for $k=0$. $$\sum_{k=0}^{0} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k = \frac{f^{(0)}(0)}{0!}(x)^0=a_0 $$
Now assume that it is true for $k=1$
Have to show that it is true for $k=k+1$ $$\sum_{k=0}^{k+1} \frac{f^{(k)}(0)}{k!}(x)^k = \sum_{k=0}^{k} \frac{f^{(k)}(0)}{k!}(x)^k + a_{k+1}x^{k+1}$$
Where to I go from here in the induction step, and is this the right way to find the Taylor Series?