<s>[INST] Need help to solve taylor series of $e^{\sin x}$ How to derive the taylor series of $e^{\sin x}$, up to $x^5$?
i just don't know how to get the answer
$$f(x) = 1 + x + \frac{x^2}{2} - \frac{x^4}{8} -\frac{x^5}{15}$$
really need some help. Thanks
 [/INST] If
$$
f(x) = a_1x + \frac{a_2}{2}x^2 + \frac{a_3}{6} x^3 + \frac{a_4}{24} x^4 + \cdots + \frac{a_n}{n!} x^n + \cdots
$$
then
$$
e^{f(x)} = 1 + a_1 x + \frac{a_2+a_1^2}{2!} + \frac{a_3 + 3a_2a_1+a_1^3}{6} x^3 + \frac{a_4+4a_3a_1 + 3a_2^2 + 6a_2a_1 + a_1^4}{24} x^4 + \cdots 
$$
The pattern is this: Consider the last case shown above, where $n=4$.  There are several ways to partition the number $4$:
$$
\begin{array}{r|c}
\text{integer partition} & \text{number of set partitions} \\
\hline
4 & 1 \\
3+1 & 4 \\
2+2 & 3 \\
2+1+1 & 6 \\
1+1+1+1 & 1
\end{array}
$$
The numbers of set partitions are the coefficients.  See this page: http://en.wikipedia.org/wiki/Exponential_formula
Now try this with
$$
a_1 = 1,\quad a_2=0,\quad a_3=-1,\quad a_4=0,\quad a_5=1,\quad\ldots\ldots
$$
$$
e^{\sin x} = 1 + x + \frac 1 2 x^2 + \frac{(-1)+0+1^3}{3} x^3 + \cdots 
$$
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