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 $f$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 105 / 1 / a / Solution

{wiki}

For a nonnegative integer $n$,
$$
(x_1+\cdots+x_r)^n
=\sum_{\alpha_1+\cdots+\alpha_r=n}
\frac{n!}{\alpha_1!\cdots\alpha_r!}
x_1^{\alpha_1}\cdots x_r^{\alpha_r}.
$$


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 Incoming links: Multinomial theorem