Compute the coefficients a_k of the polynomial

Compute the coefficients $a_k$ of the polynomial $P_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ using its successive derivatives at $x=0$. Find the degree-$n$ Maclaurin polynomial for $P_n(x)$.

Step 1: Successive derivatives at $x=0$

Differentiating repeatedly and setting $x=0$ kills every term except the one whose power exactly cancels the differentiation: \[P_n(0)=a_0, \qquad P_n'(0)=a_1, \qquad P_n''(0)=2!\,a_2, \qquad P_n'''(0)=3!\,a_3, \tag{1}\] and in general, for $0\le k\le n$: $P_n^{(k)}(0)=k!\,a_k$.

Step 2: Solve for the coefficients

\[a_k = \frac{P_n^{(k)}(0)}{k!}, \qquad k=0,1,\ldots,n\]

For $k>n$, all derivatives vanish identically.

Step 3: The Maclaurin polynomial of $P_n$ is $P_n$ itself

Applying the general degree-$n$ Maclaurin formula to $f=P_n$ and substituting the coefficients found above: \[\sum_{k=0}^n\frac{P_n^{(k)}(0)}{k!}x^k = \sum_{k=0}^n a_k x^k = P_n(x). \tag{2}\]

Step 4: Verification

\[P_2(x)=3+2x-5x^2\text{: }P_2(0)=3=a_0,\ P_2'(0)=2=a_1,\ P_2''(0)/2!=-10/2=-5=a_2\ \checkmark\]