# Taylor series expansion $(1-\varepsilon)^{-(\kappa-1)/\kappa}$

Can please someone help with finding the Taylor series expansion to this exponential? I am only looking for the first two terms, thanks!

$$(1-\varepsilon)^{-(\kappa-1)/\kappa}$$

• What's the $\&$ symbol? And what's the variable? Mar 25, 2020 at 20:01
• @Andrei sorry there is no &, I just edited it, $\kappa$ is just a constant in this case. Mar 25, 2020 at 20:03
• if $\kappa$ is a constant, then $\exp(-(\kappa-1)/\kappa)$ is also a constant Mar 25, 2020 at 20:06
• @Andrei exactly. Mar 25, 2020 at 20:07

For a constant $$C$$ and small $$\varepsilon$$, $$(1-\varepsilon)^C\approx 1-C\varepsilon+\frac {C(C-1)}2\varepsilon^2+...$$ Just plug in $$C=-\frac{\kappa-1}\kappa$$
• Take the derivatives: $f(x)=(1+x)^p$. Then $f'(x)=p(1+x)^{p-1}$ and $f''(x)=p(p-1)(1+x)^{p-2}$ Mar 25, 2020 at 20:27