= Solution
Write $S=x_1+\cdots+x_n$. The <Taylor series> of the <cosine> is
$$
\cos S=\sum_{k=0}^{\infty}\frac{(-1)^kS^{2k}}{(2k)!}.
$$
After expanding each power by the <multinomial theorem>, the coefficient of $x^\alpha$ has absolute value $1/\alpha!$ when $|\alpha|$ is <even> and is zero when $|\alpha|$ is <odd>. The <hyperbolic cosine>
$$
g(x)=\cosh(x_1+\cdots+x_n)
=\sum_{k=0}^{\infty}\frac{(x_1+\cdots+x_n)^{2k}}{(2k)!}
$$
therefore has exactly the absolute values of the coefficients of $f$. Thus $g$ is an entire <majorant series> for $f$.
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