I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions:
a) $e^{z\sin z}$
b)$(1+z)^z = e^{z \ln (1+z)}$
c)$\cos (1 + z^3) $
d) $e^{e^z}$
Although it's possible to continuously take derivatives of these functions, I feel that substitution would be a lot quicker and less painful. For a), should I use the series for $e^z$, and make this substitution:
$$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!} $$ $$e^{z\sin z} = \sum_{n=0}^\infty \frac{(z\sin z)^n}{n!}$$
For b), if my logic is correct for a), this can be solved in a similar manner, although the $\ln(1+z)$ term seems interesting, as this has its own pretty well known MacLaurin series.
For c), the substitution is $$\cos z = \sum_{n=0} ^ \infty (-1)^n \frac{z^{2n}}{(2n)!}$$ $$\cos (1+z^3) = \sum_{n=0} ^ \infty (-1)^n \frac{(1+z^3)^{2n}}{(2n)!}$$
And likewise, for d), I need to do
$$ e^{e^z} = \sum_{n=0}^\infty \frac{(e^z)^n}{n!} $$
Is this an appropriate way to go about this, or is there something else I'm missing that can really help?
