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**chisigma** Setting $\displaystyle f(x)=e^{\sin x}$ You can observe that is...

$\displaystyle \displaystyle \frac{d}{dx} \ln f(x)= \frac{f^{'}(x)}{f(x)}= \cos x$ (1)

Now if is...

$\displaystyle \displaystyle f(x)= \sum_{n=0}^{\infty} a_{n}\ x^{n}$ (2)

... the (1) becomes...

$\displaystyle \displaystyle a_{1} + 2\ a_{2}\ x + 3\ a_{3}\ x^{2} + 4\ a_{4}\ x^{3} + ... =$

$\displaystyle \displaystyle = (a_{0} + a_{1}\ x + a_{2}\ x^{2} + a_{3}\ x^{3} + ...)\ (1-\frac{x^{2}}{2} + ...)= $

$\displaystyle \displaystyle = a_{0} + a_{1}\ x + (a_{2}-\frac{1}{2})\ x^{2} - \frac{a_{1}}{2}\ x^{3}+ ...$ (3)

... and from (3) You derive...

$\displaystyle \displaystyle a_{0}=1$

$\displaystyle \displaystyle a_{1}=a_{0}=1$

$\displaystyle \displaystyle a_{2}= \frac{a_{1}}{2}= \frac{1}{2}$

$\displaystyle \displaystyle a_{3}= \frac{1}{3}\ (a_{2}-\frac{1}{2}) = \frac{1}{6}$

$\displaystyle \displaystyle a_{4}= -\frac{a_{1}}{12}= -\frac{1}{12}$

... so that is...

$\displaystyle \displaystyle e^{\sin x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + O(x^{4})$ (4)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$