Series with factorials

$\begin{aligned} \text{X} & = 1 + \dfrac{2x}{1!} - \dfrac{4x^2}{2!} - \dfrac{8x^3}{3!} + \dfrac{16x^4}{4!} + \dfrac{32x^5}{5!} - \dfrac{64x^6}{6!} - \dfrac{128x^7}{7!} + \cdots \\ \\ & = 1 - \dfrac{{(2x)}^2}{2!} + \dfrac{{(2x)}^4}{4!} - \dfrac{{(2x)}^6}{6!} + \cdots \\ & \ \ \ \ + \dfrac{2x}{1!} - \dfrac{{(2x)}^3}{3!} + \dfrac{{(2x)}^5}{5!} - \dfrac{{(2x)}^7}{7!} + \cdots \\ \\ & = \sin{(2x)} + \cos{(2x)} \ \color{#20A900}{\text{(Taylor series expansion)}} \\ \\ \therefore \lfloor \text{max(100.X)}\rfloor & = \lfloor 100\sqrt{2} \rfloor \\ \\ & = \color{#3D99F6}{\boxed{141}} \end{aligned}$

Given that:

$\begin{aligned} X & = \sum_{n=0}^\infty \frac {(2x)^n(-1)^{\left\lfloor \frac n2\right\rfloor}}{n!} = 1 + \frac {2x}{1!} - \frac {(2x)^2}{2!} - \frac {(2x)^3}{3!} + \frac {(2x)^4}{4!} + \cdots \end{aligned}$

Consider:

$\begin{aligned} Y & = \sum_{n=0}^\infty \frac {(2xi)^n}{n!} = 1 + \frac {2x}{1!}i - \frac {(2x)^2}{2!} - \frac {(2x)^3}{3!}i + \frac {(2x)^4}{4!} + \cdots = \color{#3D99F6} e^{2xi} = \cos (2x) + i \sin (2x) & \small \color{#3D99F6} \text{By Euler's formula} \end{aligned}$

We note that $X = \Re(Y) + \Im (Y) = \cos (2x) + \sin (2x)$ , where $\Re(z)$ and $\Im(z)$ are the real and imaginary parts of complex number $z$ respectively. Since $X = \cos (2x) + \sin (2x) = \sqrt 2 \sin \left(2x + \frac \pi 4\right)$ . Since $\max \left(\sin \left(2x + \frac \pi 4\right)\right) = 1$ , $\max (X) = \sqrt 2$ and $\lfloor \max(100X ) \rfloor = \boxed{141}$ .