Deduce, the Taylor series expansion of $\arcsin x$ from the Taylor Series expansion of $e^{a\arcsin x}$

Find the Taylor Series expansion of $$e^{a\arcsin x}$$ and hence deduce, the Taylor series expansion of $$\arcsin x$$.

I could find the Taylor Series expansion of $$e^{a\arcsin x}$$ as $$1+ax+\frac{a^2x^2}{2}+\frac{a^3+a}{6}x^3+\cdots$$

However, I have no idea how to deduce the series for $$\arcsin x$$ from the above expansion. I tried writing, $$e^{a\arcsin x}$$ as $$1+a\arcsin x+\frac{(a\arcsin x)^2}{2!}+\cdots$$ and comparing the coefficients with the one above, but it was not of any help because , I only got the inference that, $$x=\arcsin x$$ which was quite obvious. Any help regarding solving this issue will be highly appreciated.

• coefficient of $a$. Commented Sep 24, 2023 at 5:45
• Take the logarithm and continue with Taylor series. Commented Sep 24, 2023 at 9:19
• @ClaudeLeibovici I tried it, but it doesn't seem to help me in any way. Am I missing something? Commented Sep 24, 2023 at 12:25
• @ThomasFinley You have the Taylor series for $\exp(a\arcsin x)$, which you can regard as a series in $a$ with coefficient being a series in $x$. So look the coefficient of $a$, which gives you $\arcsin x=x+\frac16 x^3+\dots$ Commented Sep 24, 2023 at 12:54
• What is $\frac{\partial}{\partial a}e^{a\arcsin(x)}$ at $a=0$? Apply that to the power series you've found.
– robjohn
Commented Sep 24, 2023 at 14:15