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Why is this power series an asymptotic series?
I have the following series
$
S(z)= \frac{1}{2}z^2 - \frac{1}{4}z^4+\frac{1}{6}z^6 - \frac{1}{8}z^8+\cdots
$
and I want to show that it is an asymptotic series in the sense that it converge to a number and not a function. I am having problems trying to understand the reasoning behind this. The term $ \frac{1}{2}z^2$ looks somewhat strange to me. Could anyone please clarify this and what I should do to show that the term $ \frac{1}{2}z^2$ is an asymptotic series?
A:
If $c=\frac{1}{2}$ we have that
$\lim_{n \rightarrow \infty} \frac{1}{2}z^{2n} = \lim_{n \rightarrow \infty}\frac{1}{2}c^n z^{2n} = \frac{1}{2}z^2$
This is what is called the first kind of asymptotic expansion.
For the second kind we need something like $z^2-z^4$:
$\lim_{n \rightarrow \infty} (z^2-z^4)^{1/n} = \lim_{n \rightarrow \infty}(1-z)^{1/n} = e^{ -1}$
In this case:
$\lim_{n \rightarrow \infty 0b46394aab
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