Is it possible to prove the Taylor Series by mathematical induction?

Question

Originally we have: $$\displaystyle f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}{(a)}}{n!} \cdot(x-a)^n$$ So I wonder if I can write it like this: $$\displaystyle P_n(x)=\sum_{p=0}^{n} \frac{f^{(p)}{(a)}}{p!} \cdot(x-a)^p$$ Therefore, I must prove by induction that: $$\displaystyle P_{n+1}(x)=\sum_{p=0}^{n+1} \frac{f^{(p)}{(a)}}{p!} \cdot(x-a)^p$$ However, should I include the rest of the series, or does this not change the inductive step...