Chapter 11.7, Problem 76E

To determine

To prove: That $\frac{d^j}{d{x}^j}\left(\frac{{\left(x-a\right)}^k}{k!}\right)=\frac{k\left(k-1\right)\cdots \left(k-j+1\right){\left(x-a\right)}^{k-j}}{k!}$ and $\frac{d^j}{d{x}^j}{\left(\frac{{\left(x-a\right)}^k}{k!}\right)|}_{x=a}=\{\begin{array}{l}1\text{ for }k=j\\ 0\text{ for }k\ne j\end{array}$ by mathematical induction. Also, use it to prove that $T_n$ agree with $f$ at $x=a$ .