In a multichoice online test that I did the other day, I was required to select the Maclaurin series for $e^{\tan(x)}$. It was necessary for me to find the first four terms in order to establish which answer was correct. In the end of year exam, I will have reference to a Useful Information booklet (this contains a generalized Taylor polynomial approximation and Maclaurin series of $e^x$, $(1+x)^n$, $\sin(x)$, $\cos(x)$ and $\ln(1+x)$), and no calculator – hence throughout all of of my work, including online tests (which do contribute to my grade), I choose only to work with this resource, as preparation for this exam.
As my approach for this problem, I used the generalized Taylor polynomial approximation to find the Maclaurin series for $\tan(x)$ and substituted this series in place of $x$ into the given Maclaurin series for $e^x$, and double-checked my answer once I had finished every other question by using the generalized Taylor polynomial approximation to find the Maclaurin series for $e^{\tan(x)}$. Obviously, as you can imagine, both of these methods were very time consuming (especially when you consider that the other nineteen questions in the test collectively took me less than ten minutes to answer).
I'm probably missing a simple concept here. Can you please help me to establish a more elegant approach to this problem?
The choices I was given were as follows:
Best Answer
Try the following. If you only want the first four terms then you can compute everything $\bmod x^4$. Then
\begin{eqnarray*} \tan x &=& \frac{\sin x}{\cos x} \\ &\equiv& \frac{x - \frac{x^3}{6}}{1 - \frac{x^2}{2}} \bmod x^4 \\ &\equiv& \left( x - \frac{x^3}{6} \right) \left( 1 + \frac{x^2}{2} \right) \bmod x^4 \\ &\equiv& x + \frac{x^3}{3} \bmod x^4. \end{eqnarray*}
Then
\begin{eqnarray*} e^{\tan x} &\equiv& e^x e^{\frac{x^3}{3} } \bmod x^4 \\ &\equiv& \left( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \right) \left( 1 + \frac{x^3}{3} \right) \bmod x^4 \\ &\equiv& 1 + x + \frac{x^2}{2} + \frac{x^3}{2} \bmod x^4. \end{eqnarray*}
This took a little under 5 minutes on computer, and by hand it probably would have been a little faster. Is that fast enough?