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!

  • $\begingroup$ Do you know the Taylor serie? I think you can show your attempts for to solve this problem. $\endgroup$
    – mathproof
    Nov 11 '20 at 5:01


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

  • $\begingroup$ 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 '20 at 6:09
  • $\begingroup$ @ndhanson3. You are perfectly correct but here the problem is around $0$. $\endgroup$ Nov 11 '20 at 8:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.