\[

a_k=\frac{f^{(k)}(c)}{k !}

\]

The steps are given as below.

\[

\begin{gathered}

f(c)=a_0 . \\

f^{\prime}(x)=a_1+2 a_2(x-c)+3 a_3(x-c)^2+\cdots+n a_n(x-c)^{n-1}

\end{gathered}

\]

So

So

\[

\begin{gathered}

f^{\prime}(c)=a_1 . \\

f^{\prime \prime}(x)=2 a_2+6 a_3(x-c)+\cdots+n(n-1) a_n(x-c)^{n-2} .

\end{gathered}

\]

\[

f^{\prime \prime}(c)=2 a_2 .

\]

Similarly

\[

f^{(k)}(x)=k ! a_k+(2 \cdots(k+1))(x-c)+(3 \cdots(k+2))(x-c)^2+\cdots+

\]

So

\[

f^{(k)}(c)=k ! a_k .

\]

Or

\[

a_k=\frac{f^{(k)}(c)}{k !}

\]