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. $\endgroup$– user798113Nov 11, 2020 at 5:01
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1 Answer
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Hint
Use the fact that $$\sqrt{1+\sin(x)}=\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)$$
answered Nov 11, 2020 at 5:16
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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. $\endgroup$ Nov 11, 2020 at 6:09
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$\begingroup$ @ndhanson3. You are perfectly correct but here the problem is around $0$. $\endgroup$ Nov 11, 2020 at 8:11