How can we expand the following equation in Taylor series?
$\frac{1}{2}$e$^{iu/\sqrt{n}}$ + $\frac{1}{2}$e$^{-iu/\sqrt{n}}$
The solution is 1 $-$ $\frac{u^2}{2n}$ + O($\frac{1}{n^{3/2}}$), but I do not quite understand how it is obtained.
The last step is to take the limit of (1 $-$ $\frac{u^2}{2n}$ + O($\frac{1}{n^{3/2}}$))$^{nt}$ when n goes to infinite, how can we obtain the limit as e$^{-\frac{u^2}{2n}nt}$ ?
First, let me answer your first question, regarding finding the Taylor expansion:
Recall $$\cos x=\frac{e^{ix}+e^{-ix}}{2}.$$ Taking $x=u/\sqrt{n}$ gives that your function is $\cos (u/\sqrt{n}).$ Now, just use the standard Taylor expansion for $\cos x.$
Now, let me answer your last question, regarding the steps in a solution that you've seen:
Remember that $$e^x=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^n.$$ You can use this and appropriate variable substitutions to get the result that you want.