<s>[INST] Verify that for $k=3/2 ,$ $\quad f_{3/2}(x)=\frac{\sin(2x)}{2x}$ This is a part of a proof I am studying on:
Let  $f_k(x) = 1 - \frac{x^2}k+\frac{x^4}{2! k(k+1)}-\frac{x^6}{3! k(k+1)(k+2)} + \cdots \qquad (k\notin\{0,-1,-2,\ldots\}) $
For $k=3/2 ,$ it's shown that $\quad f_{3/2}(x)=\frac{\sin(2x)}{2x}$  
How is it concluded this way?
 [/INST] Note that in general
$$\sin x=\sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots.$$
thus
$$\sin 2x =2x-\frac{8x^3}{6}+\frac{32x^3}{120}...$$
and
$$\frac{\sin 2x}{2x} =1-\frac{4x^2}{6}+\frac{16x^2}{120}...$$
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