Which of the following statements is false?

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(D) Consider the function $f(x) = e^x - (1 + x)$.For $x > 0$,$f'(x) = e^x - 1 > 0$,so $f(x)$ is strictly increasing.Since $f(0) = 0$,it follows that $

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$e^x  0$

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(D) Consider the function $f(x) = e^x - (1 + x)$.For $x > 0$,$f'(x) = e^x - 1 > 0$,so $f(x)$ is strictly increasing.Since $f(0) = 0$,it follows that $e^x > 1 + x$ for all $x > 0$. Thus,option $C$ is true.Consider $g(x) = x - \log(1 + x)$.For $x > 0$,$g'(x) = 1 - \frac{1}{1 + x} = \frac{x}{1 + x} > 0$,so $g(x)$ is strictly increasing.Since $g(0) = 0$,it follows that $x > \log(1 + x)$ for all $x > 0$. Thus,option $A$ is true.Consider $h(x) = \log(1 + x) - \frac{x}{1 + x}$.For $x > 0$,$h'(x) = \frac{1}{1 + x} - \frac{(1 + x) - x}{(1 + x)^2} = \frac{1}{1 + x} - \frac{1}{(1 + x)^2} = \frac{x}{(1 + x)^2} > 0$.Since $h(0) = 0$,it follows that $\log(1 + x) > \frac{x}{1 + x}$ for all $x > 0$. Thus,option $B$ is true.Finally,for $x > 0$,$e^x > 1 + x > 1$. However,$1 - x  0$. Therefore,$e^x  0$.

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Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_{e} x| - |x - 1|$:
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$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

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