Taylor series for exp(exp(x)) using just the power series for exp(x) I'm trying to figure out how to calculate the power series for exp(exp(x)) using exp(x) and then to write down the first few terms. I have the answer for the terms but I don't know how they arrived at it 
Thanks in advance
For the first few terms we have: exp(exp(x)) = e(1 + x + x^2 + . . .)

When we put $\cdots$ we'll be ignoring all terms involving $x^n$ for $n \geq 3$. We expand like so:
\begin{align*}
\exp(\exp(x)) &= \exp\left(1 + x + \frac{1}{2}x^2 + O(x^3)\right) \\
&= e \times \exp\left(x + \frac{1}{2} x^2 + O(x^3)\right) \\
&= e\left(1 + \left(x + \frac{1}{2}x^2 + O(x^3)\right) + \frac{1}{2}\left(x + \frac{1}{2}x^2 + O(x^3)\right)^2 + O(x^3) \right) \\
&=e\left(1 + x + \frac{1}{2}x^2 + \frac{1}{2}x^2 + O(x^3) \right) \\
&=e(1 + x + x^2 + O(x^3))
\end{align*}
Normally we'd need to worry about convergence of the series, but in this case the power series for the exponential function has infinite radius of convergence so everything converges.
