1. ## Power series help

I need to show that if you expand $x(t,\varepsilon)=(1-\varepsilon^2)^{-\frac{1}{2}}e^{-\varepsilon t}sin[(1-\varepsilon^2)^{\frac{1}{2}}t]$ as a power series in $\varepsilon$, you get $x(t,\varepsilon)=sint-\varepsilon t sint+O(\varepsilon^2)$

Any help is appreciated. Thanks!

2. Originally Posted by splash
I need to show that if you expand $x(t,\varepsilon)=(1-\varepsilon^2)^{-\frac{1}{2}}e^{-\varepsilon t}sin[(1-\varepsilon^2)^{\frac{1}{2}}t]$ as a power series in $\varepsilon$, you get $x(t,\varepsilon)=sint-\varepsilon t sint+O(\varepsilon^2)$

Any help is appreciated. Thanks!
Taylor series:

$
x(t,\varepsilon)=x(t,0)+\varepsilon\, \frac{\partial x}{\partial \varepsilon}(x,0)+O(\varepsilon^2)
$

RonL