Exercise 6.7.5

Question

\begin{enumerate}[label=(\alph*)] 
    \item Follow the advice in Exercise 6.6.9 to prove the Cauchy form of the remainder:
$$
E_{N}(x)=\frac{f^{(N+1)}(c)}{N !}(x-c)^{N} x
$$
for some $c$ between 0 and $x$.
\item Use this result to prove equation (1) is valid for all $x \in(-1,1)$.

\end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item See solution to Exercise 6.6.9
    \item $$E_N(x) = \frac{-x (1-x)^{-\frac{1}{2}} }{2}  \left(\prod^N_{i=1} \frac{2i-1}{2i} \right)  \left(\frac{x-c}{1-c}\right)^N$$
    For $0 < c < x < 1$, the first term is constant, the second term is less than 1, and the last term converges to 0.
 \end{enumerate}

Exercise 6.7.5

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Exercise 6.7.5

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