<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] $$\sin x \sim x - \frac{x^3}{6} + \frac{x^5}{120}$$
$$e^x \sim 1 + x + \frac{x^2}{2}  + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120}$$
so $$e^{\sin x} \sim 1 + \sin x + \frac{\sin ^2 x}{2}  + \frac{\sin ^3 x}{6} + \frac{\sin ^ 4 x}{24} + \frac{\sin ^5x}{120} $$
Now substitute the expansion of $\sin x$, and you should get to the result
(remember to eliminate all those terms that have a degree higher than 5! :-) )
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