Exercise 30: question re (iii)

Gary


The right hand side is a "precise expression" for the difference between the fourth terms in the two series and the "is approximately equal to" sign is there because the difference between the series for $\mathcal{S}_C(z)$ and $\mathcal{S}(z)$ is the "precise expression" plus terms of smaller magnitude in $(z-a)^4$ and higher powers.

We are not discarding them but just saying they are much smaller than the third term. This is because the series is of the form
$c_0+c_1(z-a)+c_2(z-a)^2+c_3(z-a)^3+...$ and because we are using the series "in the vicinity of $a$", then $h=|(z-a)|$ is "small" (<1) and so each term will be an order of magnitude smaller than the previous term:
$c_0+c_1h+c_2h^2+c_3h^3+...$

This does assume that the values of the $c_j$ are all roughly the same order of magnitude. We are also assuming that $z$ lies within the circle of convergence at $a$.

Vasco