Chapter 10.7, Problem 64E

(a)

To determine

To show: The function $g\left(x\right)={\displaystyle \sum _{n=0}^{\infty }n{c}_n{x}^{n-1}}$ converges for $-R<x<R$.

(b)

To determine

To show: The value $\frac{{\left(x+h\right)}^n-x^n}{h}=n{c}_n^{n-1}$ using Mean Value Theorem.

(c)

To determine

To show: The value of $\left|g\left(x\right)-\frac{f\left(x+h\right)-f\left(x\right)}{h}\right|=\left|{\displaystyle \sum _{n=0}^{\infty }n{c}_n\left(x^{n-1}-c_n^{n-1}\right)}\right|$.

(d)

To determine

To show: The value of $\frac{x^{n-1}-c_n^{n-1}}{x-c_n}=\left(n-1\right)d_{n-1}^{n-2}$ using Mean Value Theorem.

(e)

To determine

The reason for $\left|x-c_n\right|<h$ and $\left|d_{n-1}\right|\le \alpha =\max \left\{\left|x\right|,\left|x+h\right|\right\}$.

(f)

To determine

To show: The value of $\left|g\left(x\right)-\frac{f\left(x+h\right)-f\left(x\right)}{h}\right|\le \left|h\right|\left|{\displaystyle \sum _{n=2}^{\infty }n\left(n-1\right)c_n{\alpha }^{n-2}}\right|$.

(g)

To determine

To show: The series ${\displaystyle \sum _{n=2}^{\infty }n\left(n-1\right){\alpha }^{n-2}}$ converges for $-R<x<R$.

(h)

To determine

To show: The limit of $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=g\left(x\right)$.