How can I find the Taylor series representation for $e^{\sqrt(x)}$ with center $c=1$? I don't think it is mathematically legal to simply take the square root of $x$ in the power series for $e^{x}$, but correct me if I'm wrong.
Should I go through and take derivatives? I am a little confused how to go about this.
Expand by differentiation.
$f(x) = e^{\sqrt(x)}, f'(x)=e^{\sqrt(x)}\frac{1}{2\sqrt{x}}, ...$ to produce:
$e^{\sqrt(x)} + e^{\sqrt(x)}\frac{1}{2\sqrt{x}}(x-1) + ...$