(i) A certain physical quantity $q(x)$ can be represented by the series $\sum_{n=0}^{\infty} c_{n} x^{n}$ in $0 \leqslant x<x_{0}$, but the series diverges for $x>x_{0}$. Describe the Euler transformation to a new series which may enable $q(x)$ to be computed for $x>x_{0}$. Give the first four terms of the new series.

Describe briefly the disadvantages of the method.

(ii) The series $\sum_{1}^{\infty} c_{r}$ has partial sums $S_{n}=\sum_{1}^{n} c_{r}$. Describe Shanks' method to approximate $S_{n}$ by

$$S_{n}=A+B C^{n}$$

giving expressions for $A, B$ and $C$.

Denote by $B_{N}$ and $C_{N}$ the values of $B$ and $C$ respectively derived from these expressions using $S_{N-1}, S_{N}$ and $S_{N+1}$ for some fixed $N$. Now let $A^{(n)}$ be the value of $A$ obtained from $(*)$ with $B=B_{N}, C=C_{N}$. Show that, if $\left|C_{N}\right|<1$,

$$\sum_{1}^{\infty} c_{r}=\lim _{n \rightarrow \infty} A^{(n)}$$

If, in fact, the partial sums satisfy

$$S_{n}=a+\alpha c^{n}+\beta d^{n}$$

with $1>|c|>|d|$, show that

$$A^{(n)}=A+\gamma d^{n}+o\left(d^{n}\right)$$

where $\gamma$ is to be found.