Chapter 11, Problem 10CRQ

a.

To determine

To fill: The blank in the statement “The Taylor series of a function f at a is ${\displaystyle \sum _{n=0}^{\infty }{a}_n{\left(x-a\right)}^n}$, where $a_n=\_\_\_\_\_$”.

b.

To determine

To fill: The blank in the statement “If f has derivatives of all orders in I and $R_N\left(x\right)=f\left(x\right)-P_N\left(x\right)$, then $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(a\right)}{n!}}{\left(x-a\right)}^n$ if and only if _______. In this case, f is said to be represented by _____ _____ _____ of f at $x=a$”.