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Find the Taylor series of $f(x) =\sqrt{1 + \sin{x}}$ around $x_0= 0$

I am aware of the taylor series expansion formula, but that's where I get stuck. Any help/a "model" solution on how to solve such problems would be helpful!

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  • $\begingroup$ Do you know the Taylor serie? I think you can show your attempts for to solve this problem. $\endgroup$
    – user798113
    Nov 11, 2020 at 5:01

1 Answer 1

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Hint

Use the fact that $$\sqrt{1+\sin(x)}=\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)$$

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  • $\begingroup$ Only for "half" of the values of $x$. With this substitution, the radius of convergence can be no more than $\pi/2$ since this equality is not true outside of $[-\pi/2,\pi/2]$. $\sqrt{1+\sin(x)}=\left|\sin(\frac{x}{2})+\cos(\frac{x}{2})\right|$ is always true. $\endgroup$
    – ndhanson3
    Nov 11, 2020 at 6:09
  • $\begingroup$ @ndhanson3. You are perfectly correct but here the problem is around $0$. $\endgroup$ Nov 11, 2020 at 8:11

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